Samacheer Kalvi Class 8 Maths Solutions Chapter 1 Numbers InText Questions

Recap Exercise (Text Book Page No. 3)

 

Question 1. The simplest form of \( \frac{125}{200} \) is
Answer: To find the simplest form of \( \frac{125}{200} \), we divide both the numerator and the denominator by their greatest common divisor. We can see that both numbers are divisible by 25.
\( \frac{125}{200} = \frac{125 \div 25}{200 \div 25} = \frac{5}{8} \)
So, the simplest form is \( \frac{5}{8} \). Understanding common factors helps in quick simplification.
In simple words: To make a fraction as small as possible, divide the top and bottom numbers by the biggest number that can divide both of them evenly. For 125/200, that number is 25, which gives 5/8.

🎯 Exam Tip: Always look for the greatest common divisor (GCD) to simplify fractions in one step. If finding the GCD is hard, divide by smaller common factors repeatedly until the fraction cannot be simplified further.

 

Question 2. Which of the following is not an equivalent fraction of \( \frac{8}{12} \) ?
(A) \( \frac{2}{3} \)
(B) \( \frac{16}{24} \)
(C) \( \frac{32}{60} \)
(D) \( \frac{24}{36} \)
Answer: (C) \( \frac{32}{60} \)
Let's check each option by simplifying \( \frac{8}{12} \) to its simplest form and then comparing.
First, simplify \( \frac{8}{12} \) by dividing the numerator and denominator by their greatest common factor, which is 4.
\( \frac{8}{12} = \frac{8 \div 4}{12 \div 4} = \frac{2}{3} \)
Now, let's check the options:
(A) \( \frac{2}{3} \): This is the simplest form, so it is equivalent.
(B) \( \frac{16}{24} \): Divide both by 8: \( \frac{16 \div 8}{24 \div 8} = \frac{2}{3} \). This is equivalent.
(C) \( \frac{32}{60} \): Divide both by 4: \( \frac{32 \div 4}{60 \div 4} = \frac{8}{15} \). This is not \( \frac{2}{3} \). So, this is not an equivalent fraction.
(D) \( \frac{24}{36} \): Divide both by 12: \( \frac{24 \div 12}{36 \div 12} = \frac{2}{3} \). This is equivalent.
The fraction \( \frac{32}{60} \) is the only one not equivalent to \( \frac{8}{12} \). Equivalent fractions represent the same part of a whole, just with different numbers.
In simple words: First, simplify the fraction 8/12 to 2/3. Then, simplify each option to see if it also becomes 2/3. The fraction 32/60 simplifies to 8/15, which is not 2/3, so it is the answer.

🎯 Exam Tip: To find if fractions are equivalent, always simplify them to their lowest terms. If their simplest forms are the same, then the fractions are equivalent.

 

Question 3. Which is bigger \( \frac{8}{9} \) or \( \frac{4}{5} \) ?
Answer: To compare fractions, we need to make their denominators the same. We find the Least Common Multiple (LCM) of the denominators 9 and 5.
The LCM of 9 and 5 is \( 9 \times 5 = 45 \).
Now, we convert both fractions to have a denominator of 45:
For \( \frac{8}{9} \): Multiply numerator and denominator by 5.
\( \frac{8}{9} = \frac{8 \times 5}{9 \times 5} = \frac{40}{45} \)
For \( \frac{4}{5} \): Multiply numerator and denominator by 9.
\( \frac{4}{5} = \frac{4 \times 9}{5 \times 9} = \frac{36}{45} \)
Now we can compare \( \frac{40}{45} \) and \( \frac{36}{45} \). Since \( 40 > 36 \), it means \( \frac{40}{45} > \frac{36}{45} \).
Therefore, \( \frac{8}{9} \) is bigger than \( \frac{4}{5} \). Comparing fractions with a common denominator is straightforward.
In simple words: To see which fraction is bigger, make the bottom numbers (denominators) the same. For 8/9 and 4/5, the common bottom number is 45. This makes them 40/45 and 36/45. Since 40 is bigger than 36, 8/9 is the bigger fraction.

🎯 Exam Tip: When comparing fractions, always find a common denominator. Another quick method is cross-multiplication: multiply the numerator of the first fraction by the denominator of the second and vice versa; the larger product corresponds to the larger fraction.

 

Question 4. Add the following fractions: \( \frac{3}{5}+\frac{5}{8}+\frac{7}{10} \).
Answer: To add these fractions, we first need to find a common denominator for 5, 8, and 10. This is the Least Common Multiple (LCM).
LCM of 5, 8, 10 is found by prime factorization:
\( 5 = 5^1 \)
\( 8 = 2^3 \)
\( 10 = 2 \times 5 \)
LCM \( = 2^3 \times 5^1 = 8 \times 5 = 40 \).
Now, convert each fraction to an equivalent fraction with a denominator of 40:
\( \frac{3}{5} = \frac{3 \times 8}{5 \times 8} = \frac{24}{40} \)
\( \frac{5}{8} = \frac{5 \times 5}{8 \times 5} = \frac{25}{40} \)
\( \frac{7}{10} = \frac{7 \times 4}{10 \times 4} = \frac{28}{40} \)
Add the new fractions:
\( \frac{24}{40} + \frac{25}{40} + \frac{28}{40} = \frac{24 + 25 + 28}{40} = \frac{77}{40} \)
This can also be written as a mixed number: \( 1 \frac{37}{40} \). Finding the LCM helps simplify fraction addition.
In simple words: To add these fractions, first find a common bottom number for 5, 8, and 10, which is 40. Change each fraction to have 40 at the bottom. Then add the top numbers together: 24 + 25 + 28 = 77. So the answer is 77/40, or 1 and 37/40.

🎯 Exam Tip: Always find the LCM of the denominators when adding or subtracting fractions. This ensures you are adding parts of the same size, which is key for accurate calculation.

 

Question 5. Simplify: \( \frac{1}{8}-\left(\frac{1}{6}-\frac{1}{4}\right) \)
Answer: We need to follow the order of operations (BODMAS/PEMDAS), starting with the operations inside the parentheses.
First, simplify \( \left(\frac{1}{6}-\frac{1}{4}\right) \). Find the LCM of 6 and 4, which is 12.
\( \frac{1}{6} = \frac{1 \times 2}{6 \times 2} = \frac{2}{12} \)
\( \frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12} \)
So, \( \frac{2}{12} - \frac{3}{12} = \frac{2 - 3}{12} = \frac{-1}{12} \)
Now, substitute this back into the original expression:
\( \frac{1}{8} - \left(\frac{-1}{12}\right) = \frac{1}{8} + \frac{1}{12} \)
Next, find the LCM of 8 and 12, which is 24.
\( \frac{1}{8} = \frac{1 \times 3}{8 \times 3} = \frac{3}{24} \)
\( \frac{1}{12} = \frac{1 \times 2}{12 \times 2} = \frac{2}{24} \)
Finally, add the fractions:
\( \frac{3}{24} + \frac{2}{24} = \frac{3+2}{24} = \frac{5}{24} \)
The order of operations is crucial for correct simplification.
In simple words: First, solve the part inside the brackets: 1/6 minus 1/4. To do this, change them to 2/12 minus 3/12, which gives -1/12. Then, put this back into the main problem: 1/8 minus -1/12 becomes 1/8 plus 1/12. Change these to 3/24 plus 2/24, and the final answer is 5/24.

🎯 Exam Tip: Always remember the order of operations (BODMAS/PEMDAS) for complex expressions. Pay close attention to negative signs, as a minus sign followed by a negative number becomes a plus.

 

Question 6. Multiply: \( 2 \frac{3}{5} \) and \( 1 \frac{4}{7} \).
Answer: To multiply mixed numbers, we first need to convert them into improper fractions.
Convert \( 2 \frac{3}{5} \): Multiply the whole number by the denominator and add the numerator. Keep the same denominator.
\( 2 \frac{3}{5} = \frac{(2 \times 5) + 3}{5} = \frac{10 + 3}{5} = \frac{13}{5} \)
Convert \( 1 \frac{4}{7} \): Multiply the whole number by the denominator and add the numerator. Keep the same denominator.
\( 1 \frac{4}{7} = \frac{(1 \times 7) + 4}{7} = \frac{7 + 4}{7} = \frac{11}{7} \)
Now, multiply the improper fractions:
\( \frac{13}{5} \times \frac{11}{7} = \frac{13 \times 11}{5 \times 7} = \frac{143}{35} \)
Finally, convert the improper fraction back into a mixed number:
\( \frac{143}{35} = 4 \frac{3}{35} \) (since \( 143 \div 35 = 4 \) with a remainder of \( 3 \)). Converting to improper fractions simplifies the multiplication process.
In simple words: First, change the mixed numbers like 2 and 3/5 into improper fractions like 13/5. Do the same for 1 and 4/7, making it 11/7. Then, multiply the top numbers (13 times 11 = 143) and the bottom numbers (5 times 7 = 35). The answer is 143/35, which can be written as 4 and 3/35.

🎯 Exam Tip: Always convert mixed numbers to improper fractions before performing multiplication or division. This simplifies the calculation and avoids common errors with mixed number arithmetic.

 

Question 7. Divide \( \frac{7}{36} \) by \( \frac{35}{81} \).
Answer: To divide one fraction by another, we multiply the first fraction by the reciprocal of the second fraction.
The reciprocal of \( \frac{35}{81} \) is \( \frac{81}{35} \).
So, the division becomes:
\( \frac{7}{36} \div \frac{35}{81} = \frac{7}{36} \times \frac{81}{35} \)
Before multiplying, we can simplify by canceling common factors diagonally:
Divide 7 and 35 by 7: \( \frac{7 \div 7}{36} \times \frac{81}{35 \div 7} = \frac{1}{36} \times \frac{81}{5} \)
Divide 36 and 81 by their common factor 9: \( \frac{1}{36 \div 9} \times \frac{81 \div 9}{5} = \frac{1}{4} \times \frac{9}{5} \)
Now, multiply the simplified fractions:
\( \frac{1}{4} \times \frac{9}{5} = \frac{1 \times 9}{4 \times 5} = \frac{9}{20} \)
Understanding reciprocals is key to division of fractions.
In simple words: To divide fractions, flip the second fraction upside down (find its reciprocal) and then multiply. So, 7/36 divided by 35/81 becomes 7/36 multiplied by 81/35. After canceling common numbers from top and bottom, the answer is 9/20.

🎯 Exam Tip: Always remember that "division by a fraction" is equivalent to "multiplication by its reciprocal". Look for opportunities to cross-cancel common factors before multiplying to keep the numbers small and simplify calculations.

 

Question 8. Fill in the boxes to complete the equivalent fractions: \( \frac{28}{44} = \frac{\Box}{66} = \frac{70}{\Box} = \frac{\Box}{121} = \frac{7}{\Box} \)
Answer: We need to find the missing numerators and denominators by using the concept of equivalent fractions. We can simplify the initial fraction to find the relationship.
First, simplify \( \frac{28}{44} \): Divide both numerator and denominator by their greatest common factor, 4.
\( \frac{28}{44} = \frac{28 \div 4}{44 \div 4} = \frac{7}{11} \)
Now, we use \( \frac{7}{11} \) to find the missing parts:
1. To find the numerator for a denominator of 66:
\( \frac{7}{11} = \frac{\Box}{66} \)
Since \( 11 \times 6 = 66 \), we multiply the numerator by 6: \( 7 \times 6 = 42 \). So the fraction is \( \frac{42}{66} \).
2. To find the denominator for a numerator of 70:
\( \frac{7}{11} = \frac{70}{\Box} \)
Since \( 7 \times 10 = 70 \), we multiply the denominator by 10: \( 11 \times 10 = 110 \). So the fraction is \( \frac{70}{110} \).
3. To find the numerator for a denominator of 121:
\( \frac{7}{11} = \frac{\Box}{121} \)
Since \( 11 \times 11 = 121 \), we multiply the numerator by 11: \( 7 \times 11 = 77 \). So the fraction is \( \frac{77}{121} \).
4. To find the denominator for a numerator of 7:
\( \frac{7}{11} = \frac{7}{\Box} \)
Here, the numerator is already 7, so the denominator must be 11. So the fraction is \( \frac{7}{11} \).
Putting it all together, the completed series of equivalent fractions is:
\( \frac{28}{44} = \frac{42}{66} = \frac{70}{110} = \frac{77}{121} = \frac{7}{11} \). Equivalent fractions keep the same value.
In simple words: First, simplify 28/44 to 7/11. Then, use 7/11 to fill in the missing numbers. If the bottom number is 66 (11 times 6), the top number is 7 times 6, which is 42. If the top number is 70 (7 times 10), the bottom number is 11 times 10, which is 110. If the bottom number is 121 (11 times 11), the top number is 7 times 11, which is 77. The last one is already 7/11.

🎯 Exam Tip: When finding equivalent fractions with missing parts, first simplify the given complete fraction to its lowest terms. This makes it easier to find the multiplier or divisor needed for the other fractions.

 

Question 9. In a city \( \frac{7}{20} \) of the population are women and \( \frac{1}{4} \) are children. Find the fraction of the population of men.
Answer: Let the total population of the city be represented by the fraction 1.
The fraction of women is \( \frac{7}{20} \).
The fraction of children is \( \frac{1}{4} \).
To find the fraction of men, we subtract the sum of women and children's fractions from the total population (1).
First, add the fractions for women and children: \( \frac{7}{20} + \frac{1}{4} \)
Find a common denominator for 20 and 4, which is 20.
\( \frac{1}{4} = \frac{1 \times 5}{4 \times 5} = \frac{5}{20} \)
So, \( \frac{7}{20} + \frac{5}{20} = \frac{7+5}{20} = \frac{12}{20} \)
Now, simplify \( \frac{12}{20} \) by dividing by 4: \( \frac{12 \div 4}{20 \div 4} = \frac{3}{5} \).
This means \( \frac{3}{5} \) of the population are women or children.
To find the fraction of men, subtract this from the total population (1):
Fraction of men \( = 1 - \frac{3}{5} = \frac{5}{5} - \frac{3}{5} = \frac{5-3}{5} = \frac{2}{5} \)
Therefore, \( \frac{2}{5} \) of the population are men. All parts of a population must add up to one whole.
In simple words: The total population is 1 whole. Women are 7/20, and children are 1/4 (which is 5/20). Add them: 7/20 + 5/20 = 12/20, which simplifies to 3/5. To find the men, subtract this from 1: 1 - 3/5 = 2/5. So, 2/5 of the population are men.

🎯 Exam Tip: When dealing with fractions representing parts of a whole, remember that the sum of all parts must always equal 1. Convert all fractions to a common denominator before adding or subtracting.

 

Question 10. Represent \( \left(\frac{1}{2}+\frac{1}{4}\right) \) by a diagram.
Answer: To represent \( \left(\frac{1}{2}+\frac{1}{4}\right) \) by a diagram, we first calculate the sum.
\( \frac{1}{2} + \frac{1}{4} = \frac{2}{4} + \frac{1}{4} = \frac{3}{4} \)
So, we need to show diagrams for \( \frac{1}{2} \), \( \frac{1}{4} \), and their sum \( \frac{3}{4} \). Visualizing fractions helps understand their values and operations.

In simple words: First, add 1/2 and 1/4, which gives 3/4. Then, draw three circles. Shade half of the first circle for 1/2. Shade a quarter of the second circle for 1/4. Finally, shade three quarters of the third circle to show the sum, 3/4.

🎯 Exam Tip: When drawing diagrams for fractions, ensure the parts are equally sized. For addition, combine the shaded parts from each fraction onto a new, equally divided whole.

Try These (Text Book Page No. 3)

 

Question 1. Is the number -7 a rational number? Why?
Answer: Yes, the number -7 is a rational number. A rational number is any number that can be written as a fraction \( \frac{p}{q} \), where \( p \) and \( q \) are integers, and \( q \) is not zero. We can write -7 as \( \frac{-7}{1} \) or \( \frac{-14}{2} \). Since -7 can be expressed as a ratio of two integers with a non-zero denominator, it fits the definition. All integers are considered rational numbers.
In simple words: Yes, -7 is a rational number. This is because any number that can be written as a simple fraction (like -7/1 or -14/2) is rational.

🎯 Exam Tip: Remember the definition of a rational number: it can be expressed as \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \). All whole numbers, integers, and terminating or repeating decimals are rational.

 

Question 2. Write any 6 rational numbers between 0 and 1.
Answer: Many rational numbers exist between 0 and 1. Here are 6 examples:
\( \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6}, \frac{1}{7} \).
These fractions all have a numerator smaller than their denominator, placing their value between 0 and 1. You could also use decimals like 0.1, 0.25, 0.5, etc. There are countless rational numbers between any two distinct rational numbers.
In simple words: You can find many fractions between 0 and 1. Some examples are 1/2, 1/3, 1/4, 1/5, 1/6, and 1/7.

🎯 Exam Tip: To find rational numbers between two given numbers, convert them to fractions with a common denominator (or to decimals) and then list fractions/decimals between them. Remember there are infinitely many such numbers.

Try These (Text Book Page No. 4)

Write the Decimal Forms of the Following Rational Numbers:

 

Question 1. \( \frac{4}{5} \)
Answer: To convert \( \frac{4}{5} \) to a decimal, we can make the denominator a power of 10. We can multiply both the numerator and denominator by 20 to get 100 in the denominator.
\( \frac{4}{5} = \frac{4 \times 20}{5 \times 20} = \frac{80}{100} \)
Now, \( \frac{80}{100} \) as a decimal is 0.80. This is a terminating decimal, as the division ends. Converting the denominator to a power of 10 often makes decimal conversion easy.
In simple words: To change 4/5 into a decimal, multiply the top and bottom by 20 to get 80/100. This is equal to 0.80.

🎯 Exam Tip: To convert a fraction to a decimal, you can either divide the numerator by the denominator directly or convert the denominator to a power of 10 (like 10, 100, 1000) if possible, by multiplying both numerator and denominator by the same number.

 

Question 2. \( \frac{6}{25} \)
Answer: To convert \( \frac{6}{25} \) to a decimal, we aim to make the denominator a power of 10. We can multiply both the numerator and denominator by 4 to get 100 in the denominator.
\( \frac{6}{25} = \frac{6 \times 4}{25 \times 4} = \frac{24}{100} \)
Now, \( \frac{24}{100} \) as a decimal is 0.24. This is a terminating decimal, which means the division process finishes. Changing the denominator to 100 is a quick way to find the decimal form.
In simple words: To change 6/25 into a decimal, multiply the top and bottom by 4 to get 24/100. This is equal to 0.24.

🎯 Exam Tip: For fractions with denominators like 2, 4, 5, 8, 10, 20, 25, 50, you can often convert them to decimals easily by making the denominator 10, 100, or 1000.

 

Question 3. \( \frac{486}{1000} \)
Answer: To convert \( \frac{486}{1000} \) to a decimal, we simply place the decimal point based on the number of zeros in the denominator. Since there are three zeros in 1000, we move the decimal point three places to the left from the end of the numerator.
\( \frac{486}{1000} = 0.486 \)
This is a terminating decimal. Fractions with powers of 10 in the denominator are very straightforward to convert to decimals.
In simple words: When the bottom number is 1000, just write the top number (486) and put the decimal point three places from the right. So, 486/1000 is 0.486.

🎯 Exam Tip: For fractions with a power of 10 in the denominator (10, 100, 1000, etc.), the number of zeros tells you how many places to move the decimal point to the left in the numerator.

 

Question 4. \( \frac{1}{9} \)
Answer: To convert \( \frac{1}{9} \) to a decimal, we perform long division of 1 by 9.
\( 1 \div 9 \)
0.111...
9 ) 1.000
      -0
      ---
      10
        -9
        ---
        10
          -9
          ---
           1
The digit '1' repeats indefinitely. So, we write this as \( 0.\overline{1} \), where the bar indicates the repeating digit. This is a non-terminating but repeating decimal.
In simple words: To change 1/9 to a decimal, divide 1 by 9. The number 1 will keep repeating after the decimal point, so we write it as 0.1 with a bar over the 1, which means 0.1111...

🎯 Exam Tip: When a digit or a block of digits repeats in the decimal expansion, use a bar (vinculum) over the repeating part. This indicates a repeating decimal.

 

Question 5. \( 3 \frac{1}{4} \)
Answer: To convert the mixed number \( 3 \frac{1}{4} \) to a decimal, we can convert the fractional part \( \frac{1}{4} \) to a decimal and then add it to the whole number part (3).
To convert \( \frac{1}{4} \) to a decimal: divide 1 by 4.
\( 1 \div 4 = 0.25 \)
Now, add this decimal to the whole number 3:
\( 3 + 0.25 = 3.25 \)
Alternatively, we could first convert \( 3 \frac{1}{4} \) to an improper fraction:
\( 3 \frac{1}{4} = \frac{(3 \times 4) + 1}{4} = \frac{12 + 1}{4} = \frac{13}{4} \)
Then, perform long division of 13 by 4:
\( 13 \div 4 = 3.25 \)
Understanding how to convert mixed numbers to decimals is very useful.
In simple words: To change 3 and 1/4 to a decimal, first change 1/4 into a decimal, which is 0.25. Then, add this to the whole number 3, so you get 3.25.

🎯 Exam Tip: For mixed numbers, convert the fractional part to a decimal first, and then add it to the whole number part. This is often simpler than converting the entire mixed number to an improper fraction before dividing.

 

Question 6. \( -2 \frac{3}{5} \)
Answer: To convert the negative mixed number \( -2 \frac{3}{5} \) to a decimal, we convert the mixed number to an improper fraction and then perform division, keeping the negative sign.
First, convert \( 2 \frac{3}{5} \) to an improper fraction (ignore the negative sign for now):
\( 2 \frac{3}{5} = \frac{(2 \times 5) + 3}{5} = \frac{10 + 3}{5} = \frac{13}{5} \)
Now, divide 13 by 5:
\( 13 \div 5 = 2.6 \)
Since the original number was negative, the decimal form will also be negative.
So, \( -2 \frac{3}{5} = -2.6 \). The process is similar to positive mixed numbers, just remember the sign.
In simple words: To change -2 and 3/5 to a decimal, first change 2 and 3/5 into 13/5. Then divide 13 by 5, which is 2.6. Since the original number was negative, the answer is -2.6.

🎯 Exam Tip: When converting negative mixed numbers to decimals, temporarily ignore the negative sign, convert the positive mixed number to a decimal, and then reapply the negative sign to the final decimal value.

Try These (Text Book Page No. 6)

 

Question 1. \( \frac{7}{3}=\frac{?}{9}=\frac{49}{?}=\frac{-21}{?} \)
Answer: We need to find the missing numerators and denominators by keeping the fractions equivalent to \( \frac{7}{3} \).
1. To find the numerator for a denominator of 9:
\( \frac{7}{3} = \frac{\Box}{9} \)
Since \( 3 \times 3 = 9 \), we multiply the numerator by 3: \( 7 \times 3 = 21 \). So, \( \frac{21}{9} \).
2. To find the denominator for a numerator of 49:
\( \frac{7}{3} = \frac{49}{\Box} \)
Since \( 7 \times 7 = 49 \), we multiply the denominator by 7: \( 3 \times 7 = 21 \). So, \( \frac{49}{21} \).
3. To find the denominator for a numerator of -21:
\( \frac{7}{3} = \frac{-21}{\Box} \)
Since \( 7 \times (-3) = -21 \), we multiply the denominator by -3: \( 3 \times (-3) = -9 \). So, \( \frac{-21}{-9} \).
The complete set of equivalent fractions is \( \frac{7}{3}=\frac{21}{9}=\frac{49}{21}=\frac{-21}{-9} \). Maintaining the ratio is fundamental for equivalent fractions.
In simple words: We need to make all fractions equal to 7/3. For 9 at the bottom (3 times 3), the top must be 7 times 3, which is 21. For 49 at the top (7 times 7), the bottom must be 3 times 7, which is 21. For -21 at the top (7 times -3), the bottom must be 3 times -3, which is -9.

🎯 Exam Tip: When finding missing values in equivalent fractions, identify the relationship (multiplication or division factor) between the known parts of the complete fraction and the partial fraction. Apply this same factor to the other part.

 

Question 2. \( \frac{-2}{5}=\frac{?}{10}=\frac{6}{?}=\frac{-8}{?} \)
Answer: We need to find the missing numerators and denominators to make the fractions equivalent to \( \frac{-2}{5} \).
1. To find the numerator for a denominator of 10:
\( \frac{-2}{5} = \frac{\Box}{10} \)
Since \( 5 \times 2 = 10 \), we multiply the numerator by 2: \( -2 \times 2 = -4 \). So, \( \frac{-4}{10} \).
2. To find the denominator for a numerator of 6:
\( \frac{-2}{5} = \frac{6}{\Box} \)
Since \( -2 \times (-3) = 6 \), we multiply the denominator by -3: \( 5 \times (-3) = -15 \). So, \( \frac{6}{-15} \).
3. To find the denominator for a numerator of -8:
\( \frac{-2}{5} = \frac{-8}{\Box} \)
Since \( -2 \times 4 = -8 \), we multiply the denominator by 4: \( 5 \times 4 = 20 \). So, \( \frac{-8}{20} \).
The complete set of equivalent fractions is \( \frac{-2}{5}=\frac{-4}{10}=\frac{6}{-15}=\frac{-8}{20} \). Correctly applying negative signs is essential.
In simple words: We need to make all fractions equal to -2/5. For 10 at the bottom (5 times 2), the top must be -2 times 2, which is -4. For 6 at the top (-2 times -3), the bottom must be 5 times -3, which is -15. For -8 at the top (-2 times 4), the bottom must be 5 times 4, which is 20.

🎯 Exam Tip: When dealing with negative equivalent fractions, remember that if you multiply/divide the numerator by a positive number, you multiply/divide the denominator by the same positive number. If you multiply/divide by a negative number, do the same for the other part, to keep the overall sign of the fraction consistent.

Try These (Text Book Page No. 7)

 

Question 1. Which of the following pairs represents equivalent rational numbers?
(i) \( \frac{-6}{4}, \frac{18}{-12} \)
(ii) \( \frac{-4}{-20}, \frac{1}{-5} \)
(iii) \( \frac{-12}{-17}, \frac{60}{85} \)
Answer: To check if two rational numbers are equivalent, we can simplify them to their lowest terms and compare.
(i) For \( \frac{-6}{4} \) and \( \frac{18}{-12} \):
Simplify \( \frac{-6}{4} \): Divide numerator and denominator by 2. \( \frac{-6 \div 2}{4 \div 2} = \frac{-3}{2} \).
Simplify \( \frac{18}{-12} \): Divide numerator and denominator by 6. \( \frac{18 \div 6}{-12 \div 6} = \frac{3}{-2} = \frac{-3}{2} \).
Since both simplify to \( \frac{-3}{2} \), they are equivalent rational numbers.
(ii) For \( \frac{-4}{-20} \) and \( \frac{1}{-5} \):
Simplify \( \frac{-4}{-20} \): Divide numerator and denominator by -4. \( \frac{-4 \div (-4)}{-20 \div (-4)} = \frac{1}{5} \).
The second fraction is \( \frac{1}{-5} = \frac{-1}{5} \).
Since \( \frac{1}{5} \neq \frac{-1}{5} \), they are not equivalent rational numbers. The position of the negative sign matters.
(iii) For \( \frac{-12}{-17} \) and \( \frac{60}{85} \):
Simplify \( \frac{-12}{-17} \): The negative signs cancel out, so it's \( \frac{12}{17} \). This fraction is already in its simplest form.
Simplify \( \frac{60}{85} \): Divide numerator and denominator by their greatest common factor, 5. \( \frac{60 \div 5}{85 \div 5} = \frac{12}{17} \).
Since both simplify to \( \frac{12}{17} \), they are equivalent rational numbers.
In simple words: To check for equivalent fractions, simplify each fraction to its simplest form. If the simplest forms are the same, they are equivalent.
(i) Both -6/4 and 18/-12 simplify to -3/2, so they are equivalent.
(ii) -4/-20 simplifies to 1/5, but 1/-5 is -1/5. These are not the same, so they are not equivalent.
(iii) Both -12/-17 (which is 12/17) and 60/85 simplify to 12/17, so they are equivalent.

🎯 Exam Tip: The simplest way to determine if two rational numbers are equivalent is to reduce each fraction to its lowest terms. If the reduced fractions are identical, the original numbers are equivalent.

 

Question 2. Write in simplest form of:
(i) \( \frac{36}{-96} \)
(ii) \( \frac{-56}{-72} \)
(iii) \( \frac{27}{18} \)
Answer: To write fractions in their simplest form, we divide both the numerator and the denominator by their greatest common divisor (GCD).
(i) For \( \frac{36}{-96} \): The GCD of 36 and 96 is 12. Remember to carry the negative sign.
\( \frac{36 \div 12}{-96 \div 12} = \frac{3}{-8} = -\frac{3}{8} \). The negative sign is usually placed in front of the fraction or with the numerator.
(ii) For \( \frac{-56}{-72} \): First, the two negative signs cancel each other, making the fraction positive: \( \frac{56}{72} \). The GCD of 56 and 72 is 8.
\( \frac{56 \div 8}{72 \div 8} = \frac{7}{9} \).
(iii) For \( \frac{27}{18} \): The GCD of 27 and 18 is 9.
\( \frac{27 \div 9}{18 \div 9} = \frac{3}{2} \). This can also be written as a mixed number \( 1 \frac{1}{2} \). Simplifying fractions helps make them easier to understand.
In simple words: To simplify each fraction, divide the top and bottom numbers by their biggest common divisor.
(i) For 36/-96, divide both by 12, giving -3/8.
(ii) For -56/-72, the two minus signs cancel, making it 56/72. Divide both by 8, giving 7/9.
(iii) For 27/18, divide both by 9, giving 3/2, which is 1 and 1/2.

🎯 Exam Tip: Always divide both the numerator and denominator by their greatest common divisor to simplify a fraction to its lowest terms. Remember that two negative signs in a fraction cancel out to make a positive fraction.

 

Question 3. Mark the following rational numbers on a number line.
(i) \( \frac{-2}{3} \)
Answer: To mark \( \frac{-2}{3} \) on a number line, we first understand its value. \( \frac{-2}{3} \) is between 0 and -1. We divide the unit segment between 0 and -1 into 3 equal parts because the denominator is 3. Then, we count 2 parts from 0 towards the left (negative direction). Number lines help visualize the positions of rational numbers.

In simple words: The fraction -2/3 is between 0 and -1. Divide the space between 0 and -1 into three equal parts. Then, mark the second part from 0 towards -1.

🎯 Exam Tip: To represent a fraction \( \frac{a}{b} \) on a number line, locate the interval between two integers. Then divide this interval into 'b' equal parts and mark the 'a'-th part. For negative fractions, move to the left from zero.

 

Question 3. Mark the following rational numbers on a number line.
(ii) \( \frac{-8}{-5} \)
Answer: First, simplify the fraction \( \frac{-8}{-5} \). The two negative signs cancel out, so \( \frac{-8}{-5} = \frac{8}{5} \).
Next, convert the improper fraction \( \frac{8}{5} \) to a mixed number: \( 8 \div 5 = 1 \) with a remainder of \( 3 \), so \( \frac{8}{5} = 1 \frac{3}{5} \).
To mark \( 1 \frac{3}{5} \) on a number line, we know it is between 1 and 2. We divide the unit segment between 1 and 2 into 5 equal parts (because the denominator is 5). Then, we count 3 parts from 1 towards the right. This visual representation helps in understanding where the number lies.

In simple words: First, -8/-5 is the same as 8/5. This is equal to 1 and 3/5. To mark this, find the spot between 1 and 2 on the number line. Divide that space into five equal parts. Then, count three parts after 1 and mark it.

🎯 Exam Tip: Always simplify fractions and convert improper fractions to mixed numbers before marking them on a number line. This makes it easier to identify the correct interval and position.

 

Question 3. Mark the following rational numbers on a number line.
(iii) \( \frac{5}{-4} \)
Answer: First, write the fraction in a standard form with the negative sign in front or with the numerator: \( \frac{5}{-4} = -\frac{5}{4} \).
Next, convert the improper fraction \( -\frac{5}{4} \) to a mixed number: \( 5 \div 4 = 1 \) with a remainder of \( 1 \), so \( -\frac{5}{4} = -1 \frac{1}{4} \).
To mark \( -1 \frac{1}{4} \) on a number line, we know it is between -1 and -2. We divide the unit segment between -1 and -2 into 4 equal parts (because the denominator is 4). Then, we count 1 part from -1 towards the left (negative direction). Placing negative fractions on a number line requires careful counting to the left.

In simple words: First, 5/-4 is the same as -5/4, which is equal to -1 and 1/4. To mark this, find the spot between -1 and -2 on the number line. Divide that space into four equal parts. Then, count one part after -1 towards -2 and mark it.

🎯 Exam Tip: When the negative sign is in the denominator, move it to the numerator or in front of the fraction for clarity. Convert to a mixed number to easily locate the interval on the number line.

Think (Text Book Page No. 15)

 

Question 1. Is zero a rational number? If so, what is its additive inverse?
Answer: Yes, zero is a rational number. A rational number can be written as \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \). Zero can be written as \( \frac{0}{1} \) or \( \frac{0}{2} \) or \( \frac{0}{-5} \), etc. Since it fits this definition, zero is a rational number. The additive inverse of a number is the number that, when added to it, results in zero. For zero, its additive inverse is itself, because \( 0 + 0 = 0 \). Zero is unique in this property among all numbers.
In simple words: Yes, zero is a rational number because you can write it as a fraction, like 0/1. The additive inverse of zero is zero itself, because adding zero to zero still gives zero.

🎯 Exam Tip: Remember that zero is a rational number because it can be expressed as \( \frac{0}{q} \) for any non-zero integer \( q \). Its additive inverse is always itself, as \( 0 + (-0) = 0 \).

Think (Text Book Page No. 16)

 

Question. What is the multiplicative inverse of 1 and -1?
Answer: The multiplicative inverse (or reciprocal) of a number is the number that, when multiplied by the original number, gives a product of 1.
For 1: The multiplicative inverse of 1 is 1 because \( 1 \times 1 = 1 \).
For -1: The multiplicative inverse of -1 is -1 because \( -1 \times (-1) = 1 \).
These two numbers are special because they are their own multiplicative inverses. Multiplicative inverses are very important in division operations.
In simple words: The multiplicative inverse of a number is what you multiply it by to get 1. For the number 1, its inverse is 1 (because 1 times 1 is 1). For the number -1, its inverse is -1 (because -1 times -1 is 1).

🎯 Exam Tip: The multiplicative inverse of a number \( x \) is \( \frac{1}{x} \). Be careful with negative numbers; the inverse of a negative number is also negative. The numbers 1 and -1 are unique as they are equal to their own multiplicative inverses.

Try These (Text Book Page No. 16)

Divide

 

Question.
(i) \( \frac{-7}{3} \) by 5
(ii) 5 by \( \frac{-7}{3} \)
(iii) \( \frac{-7}{3} \) by \( \frac{35}{6} \)
Answer: To divide by a number or a fraction, we multiply by its reciprocal.
(i) Divide \( \frac{-7}{3} \) by 5:
We can write 5 as \( \frac{5}{1} \). The reciprocal of \( \frac{5}{1} \) is \( \frac{1}{5} \).
\( \frac{-7}{3} \div 5 = \frac{-7}{3} \times \frac{1}{5} = \frac{-7 \times 1}{3 \times 5} = \frac{-7}{15} \).
(ii) Divide 5 by \( \frac{-7}{3} \):
The reciprocal of \( \frac{-7}{3} \) is \( \frac{3}{-7} \). We can write 5 as \( \frac{5}{1} \).
\( 5 \div \frac{-7}{3} = \frac{5}{1} \times \frac{3}{-7} = \frac{5 \times 3}{1 \times (-7)} = \frac{15}{-7} = -\frac{15}{7} \).
This can also be written as a mixed number: \( -2 \frac{1}{7} \).
(iii) Divide \( \frac{-7}{3} \) by \( \frac{35}{6} \):
The reciprocal of \( \frac{35}{6} \) is \( \frac{6}{35} \).
\( \frac{-7}{3} \div \frac{35}{6} = \frac{-7}{3} \times \frac{6}{35} \)
Now, we can cross-cancel common factors:
Divide -7 and 35 by 7: \( \frac{-7 \div 7}{3} \times \frac{6}{35 \div 7} = \frac{-1}{3} \times \frac{6}{5} \)
Divide 3 and 6 by 3: \( \frac{-1}{3 \div 3} \times \frac{6 \div 3}{5} = \frac{-1}{1} \times \frac{2}{5} \)
Multiply the simplified fractions:
\( \frac{-1 \times 2}{1 \times 5} = \frac{-2}{5} \).
Dividing rational numbers involves simple multiplication with the reciprocal.
In simple words: To divide, always multiply the first number by the flipped version (reciprocal) of the second number.
(i) To divide -7/3 by 5 (or 5/1), multiply -7/3 by 1/5, which gives -7/15.
(ii) To divide 5 (or 5/1) by -7/3, multiply 5/1 by 3/-7, which gives 15/-7 or -15/7.
(iii) To divide -7/3 by 35/6, multiply -7/3 by 6/35. After simplifying (canceling common numbers), you get -2/5.

🎯 Exam Tip: Remember the rule: "Keep, Change, Flip." Keep the first fraction as it is, change the division sign to multiplication, and flip the second fraction (use its reciprocal). Then multiply and simplify.

Try These (Text Book Page No. 20)

 

Question. The closure property on integers holds for subtraction and not for division. What about rational numbers? Verify.
Answer: The closure property states that if you perform an operation on two numbers from a set, the result will also be in that set.
1. **For Subtraction of Rational Numbers:**
Let's take two rational numbers, say 0 and \( \frac{1}{2} \). Both are rational numbers.
\( 0 - \frac{1}{2} = -\frac{1}{2} \).
\( -\frac{1}{2} \) is also a rational number.
If we take any two rational numbers \( a \) and \( b \), then \( a - b \) will always be a rational number. Therefore, the closure property holds for subtraction of rational numbers.
2. **For Division of Rational Numbers:**
Let's take two rational numbers, say \( \frac{5}{2} \) and 0. Both are rational numbers.
\( \frac{5}{2} \div 0 \). Division by zero is undefined.
Since the result is not a rational number, the closure property does not hold for division of rational numbers. The only exception is when the divisor is not zero. This property is important for understanding number systems.
In simple words: Closure property means if you do a math operation on two numbers from a group, the answer stays in that same group.
For subtraction of rational numbers: If you subtract one rational number from another, the answer is always a rational number. So, it holds true.
For division of rational numbers: If you try to divide a rational number by zero, the answer is not a number, so it's not a rational number. Therefore, the closure property does not hold for division.

🎯 Exam Tip: Remember that division by zero is undefined, which is the key reason why the closure property does not hold for division for most number sets (rational numbers, real numbers). Subtraction, however, usually maintains closure for integers and rational numbers.

Try These (Text Book Page No. 22)

 

Question 1. (i) Is \( \frac{3}{5}-\frac{7}{8}=\frac{7}{8}-\frac{3}{5} \) ?
Answer: We need to check if the commutative property holds for subtraction of rational numbers.
Let's calculate the Left Hand Side (LHS):
\( LHS = \frac{3}{5} - \frac{7}{8} \)
Find the LCM of 5 and 8, which is 40.
\( \frac{3}{5} = \frac{3 \times 8}{5 \times 8} = \frac{24}{40} \)
\( \frac{7}{8} = \frac{7 \times 5}{8 \times 5} = \frac{35}{40} \)
\( LHS = \frac{24}{40} - \frac{35}{40} = \frac{24 - 35}{40} = \frac{-11}{40} \)
Now, calculate the Right Hand Side (RHS):
\( RHS = \frac{7}{8} - \frac{3}{5} \)
Using the same common denominator, 40:
\( RHS = \frac{35}{40} - \frac{24}{40} = \frac{35 - 24}{40} = \frac{11}{40} \)
Since \( LHS = \frac{-11}{40} \) and \( RHS = \frac{11}{40} \), we can see that \( LHS \neq RHS \).
Therefore, \( \frac{3}{5}-\frac{7}{8} \neq \frac{7}{8}-\frac{3}{5} \). This means that subtraction of rational numbers is not commutative. The order of numbers in subtraction changes the result.
In simple words: To check if 3/5 - 7/8 is the same as 7/8 - 3/5, we calculate both sides. 3/5 - 7/8 becomes -11/40. 7/8 - 3/5 becomes 11/40. Since -11/40 is not equal to 11/40, subtraction for rational numbers is not commutative, meaning the order matters.

🎯 Exam Tip: The commutative property (a - b = b - a) generally does not hold true for subtraction. Always verify by calculating both sides of the equation if you are unsure about a property for a specific operation or number set.

 

Question 1. (ii) Is \( \frac{3}{5} \div \frac{7}{8}=\frac{7}{8} \div \frac{3}{5} \)? So, what do you conclude?
Answer: We need to check if the commutative property holds for division of rational numbers.
Let's calculate the Left Hand Side (LHS):
\( LHS = \frac{3}{5} \div \frac{7}{8} \)
To divide, we multiply by the reciprocal of the second fraction:
\( LHS = \frac{3}{5} \times \frac{8}{7} = \frac{3 \times 8}{5 \times 7} = \frac{24}{35} \)
Now, calculate the Right Hand Side (RHS):
\( RHS = \frac{7}{8} \div \frac{3}{5} \)
Multiply by the reciprocal of the second fraction:
\( RHS = \frac{7}{8} \times \frac{5}{3} = \frac{7 \times 5}{8 \times 3} = \frac{35}{24} \)
Since \( LHS = \frac{24}{35} \) and \( RHS = \frac{35}{24} \), we can see that \( LHS \neq RHS \).
Therefore, \( \frac{3}{5} \div \frac{7}{8} \neq \frac{7}{8} \div \frac{3}{5} \). We conclude that division of rational numbers is not commutative. The order of numbers in division significantly changes the outcome.
In simple words: We check if dividing 3/5 by 7/8 is the same as dividing 7/8 by 3/5. The first gives 24/35, and the second gives 35/24. Since these are not equal, division for rational numbers is not commutative, meaning the order of numbers changes the answer.

🎯 Exam Tip: Similar to subtraction, the commutative property (a ÷ b = b ÷ a) does not hold for division. Always perform the operation in the given order, as swapping the numbers will almost always lead to a different result.

Try This (Text Book Page No. 22)

 

Question. Check whether associative property holds for subtraction and division. Consider for rational numbers \( \frac{2}{3}, \frac{1}{2} \) and \( \frac{3}{4} \).
Answer: The associative property states that how numbers are grouped in an operation does not affect the result. We will test this for subtraction and division using the given rational numbers \( \frac{2}{3}, \frac{1}{2}, \frac{3}{4} \).
**1. Associative property for subtraction:**
We check if \( \left(\frac{2}{3} - \frac{1}{2}\right) - \frac{3}{4} = \frac{2}{3} - \left(\frac{1}{2} - \frac{3}{4}\right) \).
**Left Hand Side (LHS):**
First, calculate \( \left(\frac{2}{3} - \frac{1}{2}\right) \). LCM of 3 and 2 is 6.
\( \frac{2}{3} - \frac{1}{2} = \frac{4}{6} - \frac{3}{6} = \frac{1}{6} \)
Now, subtract \( \frac{3}{4} \) from this result:
\( \frac{1}{6} - \frac{3}{4} \). LCM of 6 and 4 is 12.
\( \frac{1}{6} - \frac{3}{4} = \frac{2}{12} - \frac{9}{12} = \frac{2 - 9}{12} = \frac{-7}{12} \)
So, \( LHS = \frac{-7}{12} \).
**Right Hand Side (RHS):**
First, calculate \( \left(\frac{1}{2} - \frac{3}{4}\right) \). LCM of 2 and 4 is 4.
\( \frac{1}{2} - \frac{3}{4} = \frac{2}{4} - \frac{3}{4} = \frac{2 - 3}{4} = \frac{-1}{4} \)
Now, subtract this result from \( \frac{2}{3} \):
\( \frac{2}{3} - \left(\frac{-1}{4}\right) = \frac{2}{3} + \frac{1}{4} \). LCM of 3 and 4 is 12.
\( \frac{2}{3} + \frac{1}{4} = \frac{8}{12} + \frac{3}{12} = \frac{8 + 3}{12} = \frac{11}{12} \)
So, \( RHS = \frac{11}{12} \).
Since \( LHS \neq RHS \), the associative property does not hold for subtraction of rational numbers.

**2. Associative property for division:**
We check if \( \left(\frac{2}{3} \div \frac{1}{2}\right) \div \frac{3}{4} = \frac{2}{3} \div \left(\frac{1}{2} \div \frac{3}{4}\right) \).
**Left Hand Side (LHS):**
First, calculate \( \left(\frac{2}{3} \div \frac{1}{2}\right) \). Multiply by the reciprocal:
\( \frac{2}{3} \div \frac{1}{2} = \frac{2}{3} \times \frac{2}{1} = \frac{4}{3} \)
Now, divide this result by \( \frac{3}{4} \):
\( \frac{4}{3} \div \frac{3}{4} \). Multiply by the reciprocal:
\( \frac{4}{3} \times \frac{4}{3} = \frac{16}{9} \)
So, \( LHS = \frac{16}{9} \).
**Right Hand Side (RHS):**
First, calculate \( \left(\frac{1}{2} \div \frac{3}{4}\right) \). Multiply by the reciprocal:
\( \frac{1}{2} \div \frac{3}{4} = \frac{1}{2} \times \frac{4}{3} = \frac{4}{6} = \frac{2}{3} \)
Now, divide \( \frac{2}{3} \) by this result:
\( \frac{2}{3} \div \frac{2}{3} = 1 \)
So, \( RHS = 1 \).
Since \( LHS \neq RHS \), the associative property does not hold for division of rational numbers. Grouping changes the result significantly for both subtraction and division of rational numbers.
In simple words: The associative property means you get the same answer no matter how you group the numbers (like using brackets).
For subtraction: We checked if (2/3 - 1/2) - 3/4 is the same as 2/3 - (1/2 - 3/4). The first gives -7/12, and the second gives 11/12. Since they are different, subtraction is not associative.
For division: We checked if (2/3 ÷ 1/2) ÷ 3/4 is the same as 2/3 ÷ (1/2 ÷ 3/4). The first gives 16/9, and the second gives 1. Since they are different, division is not associative.

🎯 Exam Tip: The associative property (a op (b op c) = (a op b) op c) generally holds for addition and multiplication but not for subtraction and division. Always perform calculations step-by-step according to the grouping provided, especially when the property doesn't apply.

Try This (Text Book Page No. 25)

Observe That,

 

Question. Use your reasoning skills, to find the sum of the first 7 numbers in the pattern given above: \( \frac{1}{1 \times 2}+\frac{1}{2 \times 3}=\frac{2}{3} \), \( \frac{1}{1 \times 2}+\frac{1}{2 \times 3}+\frac{1}{3 \times 4}=\frac{3}{4} \), \( \frac{1}{1 \times 2}+\frac{1}{2 \times 3}+\frac{1}{3 \times 4}+\frac{1}{4 \times 5}=\frac{4}{5} \)
Answer: Let's observe the given pattern:
For 2 numbers: \( \frac{1}{1 \times 2}+\frac{1}{2 \times 3}=\frac{2}{3} \)
For 3 numbers: \( \frac{1}{1 \times 2}+\frac{1}{2 \times 3}+\frac{1}{3 \times 4}=\frac{3}{4} \)
For 4 numbers: \( \frac{1}{1 \times 2}+\frac{1}{2 \times 3}+\frac{1}{3 \times 4}+\frac{1}{4 \times 5}=\frac{4}{5} \)
The pattern shows that the sum of 'n' fractions of the form \( \frac{1}{k \times (k+1)} \) is always \( \frac{n}{n+1} \).
So, for the sum of the first 7 numbers in this pattern, \( n=7 \).
The sum will be \( \frac{7}{7+1} = \frac{7}{8} \). Recognizing patterns simplifies complex sums.
In simple words: Look at the examples given. If you add 2 numbers, the answer is 2/3. If you add 3 numbers, the answer is 3/4. If you add 4 numbers, the answer is 4/5. The pattern is always "number of terms / (number of terms + 1)". So, for 7 numbers, the sum will be 7/8.

🎯 Exam Tip: When given a series with a clear pattern, try to find a general rule (formula) that connects the number of terms to the sum. This often involves looking at how the numerator and denominator change with each step.

Think (Text Book Page No. 26)

 

Question 1. Is the square of a prime number, prime?
Answer: No, the square of a prime number is not prime. A prime number is a whole number greater than 1 that has exactly two distinct positive divisors: 1 and itself. For example, 2, 3, 5, 7 are prime numbers.
Let \( P \) be any prime number. Its divisors are 1 and \( P \).
The square of this prime number is \( P^2 \).
The divisors of \( P^2 \) are 1, \( P \), and \( P^2 \).
Since \( P^2 \) has three distinct divisors (1, \( P \), and \( P^2 \)), it has more than two divisors. Therefore, \( P^2 \) is not a prime number. For example, the square of 2 is 4 (divisors: 1, 2, 4), which is not prime. Understanding definitions is crucial here.
In simple words: No, the square of a prime number is not prime. A prime number has only two factors (1 and itself). But a prime number squared, like \( P^2 \), will always have at least three factors: 1, P, and \( P^2 \). Since it has more than two factors, it cannot be prime.

🎯 Exam Tip: Remember the definition of a prime number (only 1 and itself as factors). The square of any prime number will always have at least three factors (1, the prime number itself, and its square), making it a composite number, not a prime number.

 

Question 2. Will the sum of two perfect squares always be a perfect square? What about their difference and their product?
Answer: Let's examine the properties of perfect squares:
**1. Sum of two perfect squares:**
The sum of two perfect squares is not always a perfect square.
Example: \( 1^2 + 2^2 = 1 + 4 = 5 \) (5 is not a perfect square).
Example: \( 3^2 + 4^2 = 9 + 16 = 25 \) (25 is a perfect square, \( 5^2 \)).
Since it's not always true, the sum is not necessarily a perfect square.
**2. Difference of two perfect squares:**
The difference of two perfect squares is not always a perfect square.
Example: \( 3^2 - 2^2 = 9 - 4 = 5 \) (5 is not a perfect square).
Example: \( 5^2 - 3^2 = 25 - 9 = 16 \) (16 is a perfect square, \( 4^2 \)).
Since it's not always true, the difference is not necessarily a perfect square.
**3. Product of two perfect squares:**
The product of two perfect squares is always a perfect square.
Let \( a^2 \) and \( b^2 \) be two perfect squares. Their product is \( a^2 \times b^2 \).
Using the property of exponents, \( a^2 \times b^2 = (a \times b)^2 \).
Since \( (a \times b)^2 \) is the square of an integer \( (a \times b) \), it is always a perfect square.
Example: \( 2^2 \times 3^2 = 4 \times 9 = 36 \) (36 is a perfect square, \( 6^2 \)).
Example: \( 4^2 \times 5^2 = 16 \times 25 = 400 \) (400 is a perfect square, \( 20^2 \)).
Understanding these properties helps in number theory problems.
In simple words:
The sum of two perfect squares (like 1+4=5) is not always a perfect square.
The difference of two perfect squares (like 9-4=5) is also not always a perfect square.
But the product of two perfect squares (like 4 times 9 = 36) is always a perfect square. This is because you can multiply their original numbers first and then square the result.

🎯 Exam Tip: For properties like "always true," "sometimes true," or "never true," use counter-examples to disprove and algebraic proofs or multiple examples to support. Remember that multiplication often preserves properties of numbers more readily than addition or subtraction.

Try These (Text Book Page No. 26)

 

Question 1. Which among 256, 576, 960, 1025, 4096 are perfect square numbers? (Hint: Try to find squares already seen).
Answer: A perfect square number is an integer that can be expressed as the product of an integer with itself. To find which numbers are perfect squares, we can try to find their square roots.
1. **256:** We know that \( 16 \times 16 = 256 \). So, \( 256 = 16^2 \). This is a perfect square.
2. **576:** We know that \( 24 \times 24 = 576 \). So, \( 576 = 24^2 \). This is a perfect square.
3. **960:** To check 960, we can look at its factors. It ends in 0, so it's divisible by 10. \( 960 = 96 \times 10 \). Neither 96 nor 10 are perfect squares, and prime factorization \( 960 = 2^5 \times 3 \times 5 \) doesn't have all exponents as even numbers. Thus, 960 is not a perfect square.
4. **1025:** A number ending in 25 might be a perfect square, but its square root must end in 5. Let's check numbers ending in 5. \( 30^2 = 900 \), \( 31^2 = 961 \), \( 32^2 = 1024 \), \( 33^2 = 1089 \). Or, \( 30^2=900, 35^2=1225 \). 1025 is not an exact square. We can see that \( 32^2 = 1024 \) and \( 33^2 = 1089 \), so 1025 is not a perfect square. Numbers ending in 5 are squares only if the digit before 5 is 2. (e.g., 25, 225, 625). Here, 1025, the digits before 25 are 10, not an even multiple of 25. The number does not fit the pattern. So, 1025 is not a perfect square.
5. **4096:** We know that \( 64 \times 64 = 4096 \). So, \( 4096 = 64^2 \). This is a perfect square.
Therefore, 256, 576, and 4096 are perfect square numbers. Checking the last digit can also give clues.
In simple words: A perfect square is a number you get by multiplying an integer by itself.
256 is 16 times 16. (Yes)
576 is 24 times 24. (Yes)
960 is not a perfect square. (No)
1025 is not a perfect square (it falls between 32x32=1024 and 33x33=1089). (No)
4096 is 64 times 64. (Yes)
So, 256, 576, and 4096 are perfect squares.

🎯 Exam Tip: To identify perfect squares, you can try to find their square roots. You can also use divisibility rules or look at the last digit (perfect squares never end in 2, 3, 7, or 8). For larger numbers, prime factorization can confirm if all prime factors have even exponents.

 

Think (Text Book Page No. 27)

 

Question 1. Is the square of a prime number, prime?
Answer: No, the square of a prime number is never prime. A prime number has only two distinct divisors: 1 and the number itself. For example, if we take the prime number 2, its square is 4. The number 4 has divisors 1, 2, and 4. Since 4 has more than two divisors, it is not a prime number. Therefore, the square of any prime number will always have at least three divisors (1, the prime number itself, and its square), meaning it is not prime. This concept is fundamental to number theory.
In simple words: No, the square of a prime number is not prime. It will have more than two factors (1, the number itself, and its square), which means it cannot be prime.

🎯 Exam Tip: Focus on the definition of a prime number having exactly two distinct factors. Any number with more than two factors is composite, not prime.

 

Question 2. Will the sum of two perfect squares always be a perfect square? What about their difference and their product? Consider the claim: 'Between the squares of the consecutive numbers n and (n + 1), there are 2n non-square numbers'. Can it be true? Find how many non-square numbers are there (i) between 4 and 9 ? (ii) between 49 and 64? and Verify the claim.
Answer:
The sum of two perfect squares is not always a perfect square. For instance, \( 1^2 + 2^2 = 1 + 4 = 5 \), which is not a perfect square. Similarly, the difference of two perfect squares is also not always a perfect square (e.g., \( 2^2 - 1^2 = 4 - 1 = 3 \)). However, the product of two perfect squares is always a perfect square (e.g., \( 2^2 \times 3^2 = 4 \times 9 = 36 = 6^2 \)). This happens because the product of powers with the same exponent can be written as the product of bases raised to that exponent.

Regarding the claim that 'Between the squares of the consecutive numbers n and (n + 1), there are 2n non-square numbers', this claim is true. Between any two consecutive perfect squares \( n^2 \) and \( (n+1)^2 \), there are exactly \( 2n \) non-square numbers. This pattern helps us quickly find how many numbers are not perfect squares in such ranges.

Therefore, we can confirm that there are \( 2n \) non-square numbers between two consecutive square numbers. This pattern is consistent for all natural numbers \( n \).
In simple words: The sum or difference of two perfect squares is not always a perfect square. But if you multiply two perfect squares, the answer is always a perfect square. Also, the claim is true: between any two perfect squares \( n^2 \) and \( (n+1)^2 \), there are \( 2n \) numbers that are not perfect squares. For example, between 4 and 9 (\( n=2 \)), there are \( 2 \times 2 = 4 \) non-square numbers (5, 6, 7, 8). Between 49 and 64 (\( n=7 \)), there are \( 2 \times 7 = 14 \) non-square numbers.

🎯 Exam Tip: When asked to verify a pattern, showing clear examples (like those for n=2 and n=7) and then stating the general rule helps score full marks.

 

Try These (Text Book Page No. 26)

 

Question 1. Which among 256, 576, 960, 1025, 4096 are perfect square numbers?
Answer: To find the perfect square numbers from the given list, we check if they can be written as an integer multiplied by itself.
* \( 256 = 16 \times 16 = 16^2 \)
* \( 576 = 24 \times 24 = 24^2 \)
* 960 is not a perfect square because \( 31^2 = 961 \) and \( 30^2 = 900 \).
* 1025 is not a perfect square because \( 32^2 = 1024 \) and \( 33^2 = 1089 \).
* \( 4096 = 64 \times 64 = 64^2 \)
Therefore, 256, 576, and 4096 are perfect square numbers. Finding the prime factors can help identify perfect squares efficiently.
In simple words: 256, 576, and 4096 are perfect squares because they are the result of multiplying a whole number by itself (like \( 16 \times 16 \), \( 24 \times 24 \), and \( 64 \times 64 \)). The other numbers are not perfect squares.

🎯 Exam Tip: For larger numbers, prime factorization or checking the last digit can quickly help rule out non-perfect squares. Numbers ending in 2, 3, 7, or 8 are never perfect squares.

 

Question 2. One can judge just by look that each of the following numbers 82, 113, 1972, 2057, 8888, 24353 is not a perfect square. Explain why?
Answer: A perfect square number can only end with certain unit digits: 0, 1, 4, 5, 6, or 9. If a number ends with any other digit (2, 3, 7, or 8), it cannot be a perfect square. This is a handy rule derived from observing the unit digits of squares of numbers from 0 to 9.
* The given numbers are 82, 113, 1972, 2057, 8888, and 24353.
* Their unit digits are 2, 3, 2, 7, 8, and 3, respectively.
* Since these unit digits are 2, 3, 7, or 8, none of these numbers can be perfect squares.
In simple words: Numbers that end in 2, 3, 7, or 8 cannot be perfect squares. All the given numbers end with these digits, so you can tell they are not perfect squares just by looking.

🎯 Exam Tip: Memorize the possible last digits of perfect squares (0, 1, 4, 5, 6, 9) to quickly identify non-perfect squares without calculation.

 

Think (Text Book Page No. 30)

 

Question. In this case, if we want to find the smallest factor with which we can multiply or divide 108 to get a square number, what should we do?
Answer: To make 108 a perfect square, we first find its prime factors. This helps identify any factors that are not in pairs.
\( 108 = 2 \times 2 \times 3 \times 3 \times 3 = 2^2 \times 3^2 \times 3^1 \)
For a number to be a perfect square, all its prime factors must appear an even number of times. Here, the factor 3 appears an odd number of times (three times), or in the grouped form, one '3' is left unpaired.
To make it a perfect square, we need to make the exponent of 3 even:
* We can multiply 108 by 3: \( 108 \times 3 = (2^2 \times 3^3) \times 3 = 2^2 \times 3^4 = (2 \times 3 \times 3)^2 = 18^2 = 324 \).
* Or, we can divide 108 by 3: \( \frac{108}{3} = \frac{2^2 \times 3^3}{3} = 2^2 \times 3^2 = (2 \times 3)^2 = 6^2 = 36 \).
Therefore, the smallest factor to either multiply or divide 108 by to get a perfect square is 3.
In simple words: First, break down 108 into its prime factors, which are \( 2 \times 2 \times 3 \times 3 \times 3 \). To be a perfect square, all prime factors must come in pairs. Since there's an extra 3, we either multiply 108 by 3 or divide 108 by 3 to make all factors pair up.

🎯 Exam Tip: Prime factorization is key for questions involving perfect squares or cubes. Look for factors that are not in pairs (for squares) or triplets (for cubes) and multiply or divide by those missing factors.

 

Try These (Text Book Page No. 32)

 

Find The Square Root By Long Division Method.

 

Question 1. 400
Answer: To find the square root of 400 using the long division method, we follow these steps:

From the long division, we find that \( \sqrt{400} = 20 \). The method helps systematically extract the square root by repeatedly finding the largest possible divisor.
In simple words: To find the square root of 400 using the long division method, we pair the digits from the right and divide step by step. The answer is 20.

🎯 Exam Tip: In the long division method for square roots, always bring down pairs of digits at each step, and ensure the divisor for each step is twice the quotient obtained so far, plus the new digit.

 

Question 2. 1764
Answer: To find the square root of 1764 using the long division method, we proceed as follows:

From the long division, we get \( \sqrt{1764} = 42 \). This method systematically finds the largest integer whose square is less than or equal to the current number at each step.
In simple words: Using the long division method, we can find that the square root of 1764 is 42.

🎯 Exam Tip: Ensure careful calculation at each step, especially when selecting the next digit for the divisor by trying digits from 1 to 9.

 

Question 3. 9801
Answer: To find the square root of 9801 using the long division method, we follow the steps below:

From the long division, we find that \( \sqrt{9801} = 99 \). This process works for any perfect square number to find its square root efficiently.
In simple words: We find that the square root of 9801 using the long division method is 99.

🎯 Exam Tip: For numbers like 9801, ending in 1 suggests the root ends in 1 or 9. This can help in checking your steps and making educated guesses for the next digit in the quotient.

 

Try These (Text Book Page No. 32)

 

Without calculating the square root, guess the number of digits in the square root of the following numbers:

 

Question 1. 14400
Answer: To find the number of digits in the square root of 14400 without calculating it, we group the digits of the number in pairs, starting from the right. A single digit or the leftmost pair of digits makes the first group.
14400 has 5 digits. Grouping them gives \( \overline{1} \overline{44} \overline{00} \). There are three groups.
The number of digits in the square root is equal to the number of groups formed. So, the square root of 14400 will have 3 digits.
(For verification, \( \sqrt{14400} = 120 \), which indeed has 3 digits). This grouping method provides a quick way to determine the size of the square root.
In simple words: For 14400, group the digits from the right: \( \overline{1} \overline{44} \overline{00} \). There are three groups, so its square root (which is 120) has 3 digits.

🎯 Exam Tip: For a perfect square with 'n' digits, its square root will have \( \frac{n}{2} \) digits if 'n' is even, and \( \frac{n+1}{2} \) digits if 'n' is odd. Here, n=5 (odd), so \( \frac{5+1}{2} = 3 \) digits.

 

Question 2. 390625
Answer: To find the number of digits in the square root of 390625 without calculating it, we group its digits in pairs from the right.
390625 has 6 digits. Grouping them gives \( \overline{39} \overline{06} \overline{25} \). There are three groups.
The number of digits in the square root is equal to the number of groups formed. So, the square root of 390625 will have 3 digits.
(For verification, \( \sqrt{390625} = 625 \), which indeed has 3 digits). This method is useful for quickly estimating the number of digits in large square roots.
In simple words: For 390625, grouping digits from the right gives 3 groups (\( \overline{39} \overline{06} \overline{25} \)). So, its square root (which is 625) has 3 digits.

🎯 Exam Tip: An even number of digits 'n' in the original number means its square root will have \( n/2 \) digits. Here, n=6 (even), so \( 6/2 = 3 \) digits.

 

Question 3. 100000000
Answer: To determine the number of digits in the square root of 100,000,000 without calculating it, we group the digits from the right in pairs.
100,000,000 has 9 digits. Grouping them gives \( \overline{1} \overline{00} \overline{00} \overline{00} \overline{00} \). There are five groups.
The number of digits in the square root is equal to the number of groups formed. So, the square root of 100,000,000 will have 5 digits.
(For verification, \( \sqrt{100000000} = 10,000 \), which indeed has 5 digits). This method provides a reliable way to predict the length of a square root.
In simple words: For 100,000,000, we group the digits from the right. There are 5 groups, so its square root (which is 10,000) has 5 digits.

🎯 Exam Tip: An odd number of digits 'n' in the original number means its square root will have \( (n+1)/2 \) digits. Here, n=9 (odd), so \( \frac{9+1}{2} = 5 \) digits.

 

Try These (Text Book Page No. 33)

 

Find The Square Root Of

 

Question 1. 5.4756
Answer: To find the square root of 5.4756 using the long division method, we follow these steps, pairing digits from the decimal point.

Thus, \( \sqrt{5.4756} = 2.34 \). When finding the square root of a decimal number, it is essential to pair digits from the decimal point outwards, adding zeros if necessary.
In simple words: Using the long division method, the square root of 5.4756 is 2.34.

🎯 Exam Tip: For decimal numbers, pair digits from the decimal point to the left for the whole part and to the right for the fractional part. Place the decimal in the quotient when you cross the decimal in the dividend.

 

Question 2. 19.36
Answer: To find the square root of 19.36 using the long division method, we perform the following steps:

Therefore, \( \sqrt{19.36} = 4.4 \). Remember to place the decimal point in the quotient directly above the decimal point in the number being divided, as soon as you move to the decimal part.
In simple words: The square root of 19.36, found using the long division method, is 4.4.

🎯 Exam Tip: Place the decimal point in the quotient directly above the decimal point in the dividend when using long division for square roots, maintaining correct alignment.

 

Question 3. 116.64
Answer: To find the square root of 116.64 using the long division method, we follow these detailed steps:

So, \( \sqrt{116.64} = 10.8 \). The long division method works consistently for numbers with both whole and decimal parts, ensuring accuracy in square root calculations.
In simple words: The square root of 116.64, calculated using long division, is 10.8.

🎯 Exam Tip: For decimal square roots, make sure to pair digits correctly on both sides of the decimal point, adding zeros to the right if needed to form pairs.

 

Think (Text Book Page No. 33)

 

Question. Try to fill in the blanks using \( \sqrt{ab} = \sqrt{a} \times \sqrt{b} \).
Answer: This table demonstrates the property that the square root of a product of two numbers is equal to the product of their individual square roots, i.e., \( \sqrt{ab} = \sqrt{a} \times \sqrt{b} \).

This table shows that the square root of a product of two numbers is equal to the product of their individual square roots. This property, \( \sqrt{ab} = \sqrt{a} \times \sqrt{b} \), makes calculations with square roots much simpler.
In simple words: The table shows that when you take the square root of two numbers multiplied together, it's the same as taking the square root of each number separately and then multiplying those results. This rule helps us simplify square root problems.

🎯 Exam Tip: This property \( \sqrt{ab} = \sqrt{a} \times \sqrt{b} \) is very useful for simplifying square roots of large numbers by factoring them into smaller perfect squares.

 

Try These (Text Book Page No. 34)

 

Question. Using this method, find the square root of the numbers 1.2321 and 11.9025.
Answer:
(i) 1.2321
To find the square root of 1.2321, we can convert it into a fraction first:
\( \sqrt{1.2321} = \sqrt{\frac{12321}{10000}} \)
We know that \( \sqrt{12321} = 111 \) and \( \sqrt{10000} = 100 \).
So, \( \sqrt{1.2321} = \frac{111}{100} = 1.11 \). This method converts the decimal to a fraction, finds the square root of the numerator and denominator separately, and then converts back to decimal, which is particularly effective for perfect squares.

(ii) 11.9025
To find the square root of 11.9025, we express it as a fraction:
\( \sqrt{11.9025} = \frac{\sqrt{119025}}{\sqrt{10000}} \)
We know that \( \sqrt{119025} = 345 \) and \( \sqrt{10000} = 100 \).
So, \( \sqrt{11.9025} = \frac{345}{100} = 3.45 \). This approach makes it easy to handle decimal square roots when the numbers involved are perfect squares.
In simple words: To find the square root of 1.2321, change it to the fraction \( \frac{12321}{10000} \), then take the square root of the top and bottom to get \( \frac{111}{100} \), which is 1.11. For 11.9025, similarly, it becomes \( \frac{119025}{10000} \), and the square root is \( \frac{345}{100} \), which is 3.45.

🎯 Exam Tip: Converting decimals to fractions before finding square roots can simplify complex calculations, especially for perfect squares, as it separates the whole number calculation.

 

Write the numbers in ascending order.

 

Question 1. 4, \( \sqrt{14} \), 5
Answer: To arrange these numbers in ascending order (from smallest to largest), we compare their values by squaring them all. This removes the square root and allows for direct comparison.
* Square of 4: \( 4^2 = 16 \)
* Square of \( \sqrt{14} \): \( (\sqrt{14})^2 = 14 \)
* Square of 5: \( 5^2 = 25 \)
Comparing the squared values in ascending order: 14, 16, 25.
So, in ascending order, the original numbers are \( \sqrt{14} \), 4, 5. Squaring helps compare numbers that include square roots by converting them to whole numbers or easily comparable forms.
In simple words: To order 4, \( \sqrt{14} \), and 5, we square each one. We get 16, 14, and 25. The smallest is 14, then 16, then 25. So, the original numbers in order are \( \sqrt{14} \), 4, 5.

🎯 Exam Tip: To compare numbers involving square roots, it is often easiest to square all numbers (or parts) to remove the square root and then compare the resulting values. This works reliably for positive numbers.

 

Question 2. 7, \( \sqrt{65} \), 8
Answer: To arrange these numbers in ascending order, we square each number to make comparison easier, especially since one number is a square root.
* Square of 7: \( 7^2 = 49 \)
* Square of \( \sqrt{65} \): \( (\sqrt{65})^2 = 65 \)
* Square of 8: \( 8^2 = 64 \)
Comparing the squared values in ascending order: 49, 64, 65.
So, in ascending order, the original numbers are 7, 8, \( \sqrt{65} \). This method is particularly useful when comparing numbers where one is an irrational square root, simplifying the process of ordering.
In simple words: To order 7, \( \sqrt{65} \), and 8, we square them. We get 49, 65, and 64. The smallest is 49, then 64, then 65. So, the original numbers in order are 7, 8, \( \sqrt{65} \).

🎯 Exam Tip: Squaring numbers for comparison only works if all numbers are positive. If negative numbers are involved, extra care is needed as squaring negative numbers changes their sign and magnitude. In this case, all numbers are positive.

 

Try These (Text Book Page No. 37)

 

Find The Ones Digit In The Cubes Of Each Of The Following Numbers.

 

Question. (i) 12
Answer: To find the ones digit of \( 12^3 \), we only need to look at the ones digit of the number 12, which is 2. The ones digit of a cube depends solely on the ones digit of the original number.
The ones digit of \( 2^3 \) is \( 2 \times 2 \times 2 = 8 \).
Therefore, the ones digit in \( 12^3 \) is 8.
In simple words: The ones digit of 12 is 2. When you cube 2, you get 8. So, the ones digit of \( 12^3 \) is 8.

🎯 Exam Tip: Memorize the ones digits of cubes from 0 to 9 to quickly find the last digit of any cube without full calculation.

 

Question. (ii) 27
Answer: To find the ones digit of \( 27^3 \), we consider the ones digit of 27, which is 7. This is a shortcut that avoids calculating the full cube.
The ones digit of \( 7^3 \) is \( 7 \times 7 \times 7 = 343 \), so the ones digit is 3.
Therefore, the ones digit in \( 27^3 \) is 3.
In simple words: The ones digit of 27 is 7. When you cube 7, the result ends in 3 (343). So, the ones digit of \( 27^3 \) is 3.

🎯 Exam Tip: The sequence of ones digits for cubes (0-9) is 0, 1, 8, 7, 4, 5, 6, 3, 2, 9, which is unique for each digit, meaning no two distinct digits (0-9) have the same unit digit for their cubes.

 

Question. (iii) 38
Answer: To find the ones digit of \( 38^3 \), we look at the ones digit of 38, which is 8.
The ones digit of \( 8^3 \) is \( 8 \times 8 \times 8 = 512 \), so the ones digit is 2.
Therefore, the ones digit in \( 38^3 \) is 2.
In simple words: The ones digit of 38 is 8. When you cube 8, the result ends in 2 (512). So, the ones digit of \( 38^3 \) is 2.

🎯 Exam Tip: Notice that for 2, the last digit of its cube is 8; for 8, it's 2. Similarly, for 3, it's 7; for 7, it's 3. These complementary pairs are useful to remember.

 

Question. (iv) 53
Answer: To find the ones digit of \( 53^3 \), we only need to look at the ones digit of 53, which is 3.
The ones digit of \( 3^3 \) is \( 3 \times 3 \times 3 = 27 \), so the ones digit is 7.
Therefore, the ones digit in \( 53^3 \) is 7.
In simple words: The ones digit of 53 is 3. When you cube 3, the result ends in 7 (27). So, the ones digit of \( 53^3 \) is 7.

🎯 Exam Tip: This quick method saves time in multiple-choice questions or when a full calculation of the cube is not required.

 

Question. (v) 71
Answer: To find the ones digit of \( 71^3 \), we focus on the ones digit of 71, which is 1.
The ones digit of \( 1^3 \) is \( 1 \times 1 \times 1 = 1 \).
Therefore, the ones digit in \( 71^3 \) is 1. Numbers ending in 1, 0, 4, 5, 6, or 9 have cubes that also end in the same digit, simplifying the prediction.
In simple words: The ones digit of 71 is 1. When you cube 1, you get 1. So, the ones digit of \( 71^3 \) is 1.

🎯 Exam Tip: For digits 0, 1, 4, 5, 6, and 9, the unit digit of their cube is the same as the digit itself. This makes these cases very easy to remember.

 

Question. (vi) 84
Answer: To find the ones digit of \( 84^3 \), we look at the ones digit of 84, which is 4.
The ones digit of \( 4^3 \) is \( 4 \times 4 \times 4 = 64 \), so the ones digit is 4.
Therefore, the ones digit in \( 84^3 \) is 4.
In simple words: The ones digit of 84 is 4. When you cube 4, the result ends in 4 (64). So, the ones digit of \( 84^3 \) is 4.

🎯 Exam Tip: The pattern of unit digits for cubes is cyclic and easy to observe, making these predictions straightforward for all numbers.

 

Try These (Text Book Page No. 41)

 

Expand The Following Numbers Using Exponents:

 

Question. (i) 8120
Answer: We can expand 8120 using powers of 10, which clearly shows the place value of each digit.
\( 8120 = (8 \times 1000) + (1 \times 100) + (2 \times 10) + (0 \times 1) \)
Using exponents for 10, this becomes:
\( 8120 = (8 \times 10^3) + (1 \times 10^2) + (2 \times 10^1) + (0 \times 10^0) \).
This expanded form is crucial for understanding how our number system works.
In simple words: To expand 8120 using exponents, we write it as \( 8 \times 10^3 + 1 \times 10^2 + 2 \times 10^1 + 0 \times 10^0 \).

🎯 Exam Tip: Remember that any non-zero number raised to the power of 0 is 1. This means \( 10^0 = 1 \).

 

Question. (ii) 20,305
Answer: We expand 20,305 by breaking it down into its place values, using powers of 10 for each position.
\( 20305 = (2 \times 10000) + (0 \times 1000) + (3 \times 100) + (0 \times 10) + (5 \times 1) \)
In exponential form, this is:
\( 20305 = (2 \times 10^4) + (0 \times 10^3) + (3 \times 10^2) + (0 \times 10^1) + (5 \times 10^0) \).
Even if a digit is zero, its corresponding power of ten is still part of the complete expanded form.
In simple words: The expanded form of 20,305 using exponents is \( 2 \times 10^4 + 0 \times 10^3 + 3 \times 10^2 + 0 \times 10^1 + 5 \times 10^0 \).

🎯 Exam Tip: Pay attention to the zero digits; they still correspond to a power of 10, even if their term simplifies to zero. Always include them for a complete expansion.

 

Question. (iii) 3652.01
Answer: To expand 3652.01 using exponents, we consider the whole number part and the decimal part separately, using positive and negative powers of 10.
\( 3652.01 = 3000 + 600 + 50 + 2 + \frac{0}{10} + \frac{1}{100} \)
Using powers of 10, this becomes:
\( 3652.01 = (3 \times 10^3) + (6 \times 10^2) + (5 \times 10^1) + (2 \times 10^0) + (0 \times 10^{-1}) + (1 \times 10^{-2}) \).
Negative exponents are used for digits after the decimal point, representing fractions of ten.
In simple words: To write 3652.01 in expanded form with exponents, we get \( 3 \times 10^3 + 6 \times 10^2 + 5 \times 10^1 + 2 \times 10^0 + 0 \times 10^{-1} + 1 \times 10^{-2} \).

🎯 Exam Tip: Remember that \( 10^{-1} = \frac{1}{10} \) and \( 10^{-2} = \frac{1}{100} \) for the decimal places. Each digit after the decimal point corresponds to a decreasing negative power of 10.

 

Question. (iv) 9426.521
Answer: We expand 9426.521 using powers of 10, including negative exponents for the decimal part, to show the value of each digit.
\( 9426.521 = (9 \times 1000) + (4 \times 100) + (2 \times 10) + (6 \times 1) + (\frac{5}{10}) + (\frac{2}{100}) + (\frac{1}{1000}) \)
In exponential form, this is:
\( 9426.521 = (9 \times 10^3) + (4 \times 10^2) + (2 \times 10^1) + (6 \times 10^0) + (5 \times 10^{-1}) + (2 \times 10^{-2}) + (1 \times 10^{-3}) \).
This expansion is crucial for understanding scientific notation and place value system.
In simple words: The expanded form of 9426.521 using exponents is \( 9 \times 10^3 + 4 \times 10^2 + 2 \times 10^1 + 6 \times 10^0 + 5 \times 10^{-1} + 2 \times 10^{-2} + 1 \times 10^{-3} \).

🎯 Exam Tip: For any number, the digit immediately to the left of the decimal point is multiplied by \( 10^0 \).

 

Try These (Text Book Page No. 42)

 

Question. Verify the following rules (as we did above). Here, a,b are non-zero integers and m are any integers:
1. Product rule of powers: \( a^m \times b^m = (ab)^m \)
2. Quotient rule of powers: \( \frac{a^m}{b^m}=\left(\frac{a}{b}\right)^m \)
3. Zero exponent rule: \( a^0 = 1 \)
Answer: We will verify these exponent rules by using specific integer values for \( a \), \( b \), and \( m \). Let's choose simple values: \( a = 2 \), \( b = 3 \), and \( m = 2 \).
* **1. Product Rule:** \( a^m \times b^m = (ab)^m \)
Left Hand Side (LHS): \( 2^2 \times 3^2 = 4 \times 9 = 36 \)
Right Hand Side (RHS): \( (2 \times 3)^2 = 6^2 = 36 \)
Since LHS = RHS (\( 36 = 36 \)), the product rule is verified. This rule simplifies multiplying powers that have the same exponent.
* **2. Quotient Rule:** \( \frac{a^m}{b^m}=\left(\frac{a}{b}\right)^m \)
Left Hand Side (LHS): \( \frac{2^2}{3^2} = \frac{4}{9} \)
Right Hand Side (RHS): \( \left(\frac{2}{3}\right)^2 = \frac{2^2}{3^2} = \frac{4}{9} \)
Since LHS = RHS (\( \frac{4}{9} = \frac{4}{9} \)), the quotient rule is verified. This rule simplifies dividing powers that have the same exponent.
* **3. Zero Exponent Rule:** \( a^0 = 1 \)
For \( a = 2 \), \( 2^0 = 1 \). This rule states that any non-zero number raised to the power of zero equals one, which is a fundamental property of exponents.
All three rules are verified with our chosen values. Using examples helps to concretely understand abstract mathematical rules.
In simple words: We can check these rules with numbers. If we take \( a=2 \), \( b=3 \), and \( m=2 \):
1. \( 2^2 \times 3^2 = 36 \), and \( (2 \times 3)^2 = 36 \). So the product rule works.
2. \( \frac{2^2}{3^2} = \frac{4}{9} \), and \( (\frac{2}{3})^2 = \frac{4}{9} \). So the quotient rule works.
3. \( 2^0 = 1 \). So the zero exponent rule works.

🎯 Exam Tip: When verifying rules, pick simple integer values for variables that avoid division by zero or overly complex calculations, making the verification clear and easy to follow.

 

Try These (Text Book Page No. 44)

 

Question 1. Write in standard form: Mass of planet Uranus is \( 8.68 \times 10^{25} \) kg.
Answer: To write \( 8.68 \times 10^{25} \) kg in standard form (also known as usual form), we move the decimal point 25 places to the right. The positive exponent indicates a large number.
Starting with 8.68, we move the decimal two places for the '68' and then add 23 more zeros.
So, \( 8.68 \times 10^{25} \) kg in standard form is 86,800,000,000,000,000,000,000,000 kg.
(This number has 23 zeros after 868). Scientific notation makes very large or very small numbers easier to write and read by compressing them.
In simple words: To write \( 8.68 \times 10^{25} \) kg in its normal, long form, we move the decimal point 25 places to the right. This means it becomes 868 followed by 23 zeros.

🎯 Exam Tip: For positive powers of 10, move the decimal point to the right. The exponent tells you the total number of places to move the decimal.

 

Question 2. Write in scientific notation:
(i) 0.000012005
(ii) 4312.345
Answer:
(i) 0.000012005
To write 0.000012005 in scientific notation, we move the decimal point to the right until there is only one non-zero digit before it. We moved the decimal point 5 places to the right to get 1.2005. Since we moved it to the right, the exponent of 10 will be negative. So, \( 0.000012005 = 1.2005 \times 10^{-5} \). This form clearly shows the magnitude and significant figures of a very small number.

(ii) 4312.345
To write 4312.345 in scientific notation, we move the decimal point to the left until there is only one non-zero digit before it. We moved the decimal point 3 places to the left to get 4.312345. Since we moved it to the left, the exponent of 10 will be positive. So, \( 4312.345 = 4.312345 \times 10^3 \). This is a compact way to represent large numbers while keeping all significant digits.
In simple words: For 0.000012005, move the decimal 5 places to the right to get 1.2005. Because you moved right, it's \( 1.2005 \times 10^{-5} \). For 4312.345, move the decimal 3 places to the left to get 4.312345. Because you moved left, it's \( 4.312345 \times 10^3 \).

🎯 Exam Tip: For numbers less than 1, the exponent in scientific notation will always be negative. For numbers 10 or greater, the exponent will be positive.

 

Question. (iii) 0.10524
Answer: To write 0.10524 in scientific notation, we move the decimal point one place to the right to get 1.0524. We moved the decimal one place to the right, so the exponent of 10 will be -1. So, \( 0.10524 = 1.0524 \times 10^{-1} \). This format clearly indicates that the original number is less than 1, using a standard scientific representation.
In simple words: For 0.10524, move the decimal 1 place to the right to get 1.0524. Since you moved right, it's \( 1.0524 \times 10^{-1} \).

🎯 Exam Tip: A negative exponent of 10 indicates that the original number was a decimal between 0 and 1. The absolute value of the exponent shows how many places the decimal was moved.