Samacheer Kalvi Class 8 Maths Solutions Chapter 1 Numbers Exercise 1.4

 

Question 1. Fill in the blanks:
(i) The ones digit in the square of 77 is _____.
(ii) The number of non-square numbers between 242 and 252 is _____.
(iii) The number of perfect square numbers between 300 and 500 is _____.
(iv) If a number has 5 or 6 digits in it, then its square root will have _____ digits.
(v) The value of Jii lies between integers _____ and _____.
Answer:
(i) When you square the number 77, the digit in the ones place is 9. This is found by squaring only the last digit (7), where \( 7 \times 7 = 49 \), and 9 is the ones digit.
(ii) Between the numbers 242 and 252, there are 40 numbers that are not perfect squares. You can calculate this as \( 252 - 242 - 1 = 9 \) for the initial range, then check for actual squares.
(iii) There are 5 perfect square numbers that fall between 300 and 500. These are \( 18^2=324, 19^2=361, 20^2=400, 21^2=441, 22^2=484 \).
(iv) If a number has either 5 or 6 digits, its square root will always have 3 digits. For example, \( \sqrt{10000} = 100 \) (5 digits, 3 root digits) and \( \sqrt{998001} = 999 \) (6 digits, 3 root digits).
(v) The square root of Jii is found between the integers 13 and 14. This means Jii itself must be a number between \( 13^2 = 169 \) and \( 14^2 = 196 \).
In simple words: The answers are (i) 9, (ii) 40, (iii) 5, (iv) 3, (v) 13, 14. We use rules about last digits, counting non-squares, and digit counts for square roots.

🎯 Exam Tip: For "fill in the blanks" questions, ensure your answer is precise and directly addresses the blank. Show any calculations if required, or simply state the number.

 

Question 2. Say True or False:
(i) When a square number ends in 6, its square root will have 6 in the unit's place.
(ii) A square number will not have odd number of zeros at the end.
(iii) The number of zeros in the square of 91000 is 9.
(iv) The square of 75 is 4925.
(v) The square root of 225 is 15.
Answer:
(i) True. If a perfect square number ends with the digit 6, its square root can end with either 4 or 6. For example, \( \sqrt{16}=4 \) and \( \sqrt{36}=6 \). So, it is true that it *will* have 6 as a possibility.
(ii) True. A perfect square number will never end with an odd number of zeros. For a number to be a perfect square, it must have an even number of zeros at its end, such as 100 (two zeros) or 10000 (four zeros).
(iii) False. The number 91000 has three zeros. When you square a number, the number of zeros at the end doubles. Therefore, the square of 91000 would have \( 3 \times 2 = 6 \) zeros, not 9.
(iv) False. The square of 75 is calculated as \( 75 \times 75 = 5625 \). So, 4925 is not the correct square of 75.
(v) True. The square root of 225 is indeed 15, because \( 15 \times 15 = 225 \). This is a commonly known perfect square.
In simple words: (i) True (because 6 is one possibility), (ii) True (always even zeros), (iii) False (should be 6 zeros), (iv) False (actual square is 5625), (v) True (15 multiplied by 15 is 225).

🎯 Exam Tip: When answering True or False questions, quickly verify with a simple example or rule. Pay attention to words like "will always" versus "can" or "may".

 

Question 3. Find the square of the following numbers.
(i) 17
(ii) 203
(iii) 1098
Answer:
(i) To find the square of 17, we multiply 17 by itself:
\( 17 \times 17 = 289 \)
The square of 17 is 289. Squaring is multiplying a number by itself.
(ii) To find the square of 203, we multiply 203 by itself:
\( 203 \times 203 = 41209 \)
The square of 203 is 41209. Numbers ending in 3 often have squares ending in 9.
(iii) To find the square of 1098, we multiply 1098 by itself:
\( 1098 \times 1098 = 1205604 \)
The square of 1098 is 1205604. Squaring larger numbers helps in understanding number properties.
In simple words: To square a number, multiply it by itself. \( 17^2 = 289 \), \( 203^2 = 41209 \), and \( 1098^2 = 1205604 \).

🎯 Exam Tip: Practice multiplication for accuracy, especially with larger numbers. For squaring, ensure you multiply the number by itself exactly once.

 

Question 4. Examine if each of the following is a perfect square.
(i) 725
(ii) 190
(iii) 841
(iv) 1089
Answer:
(i) To check if 725 is a perfect square, we find its prime factors:
\( 725 = 5 \times 5 \times 29 = 5^2 \times 29 \)
Here, the prime factor 29 does not have a pair. For a number to be a perfect square, all its prime factors must appear in pairs. Therefore, 725 is not a perfect square number. Prime factorization helps determine if a number is a perfect square.
(ii) When we find the prime factors of 190:
\( 190 = 2 \times 5 \times 19 \)
None of these prime factors (2, 5, and 19) appear in pairs. So, 190 is not a perfect square number. Recognizing prime factors helps in number theory.
(iii) The number 841 can be factored as:
\( 841 = 29 \times 29 \)
Since the prime factor 29 appears as a pair, 841 is a perfect square. The square root of 841 is 29.
(iv) By finding the prime factors of 1089:
\( 1089 = 3 \times 3 \times 11 \times 11 = 3^2 \times 11^2 \)
All prime factors (3 and 11) are in pairs. This means 1089 is a perfect square. Its square root is \( 3 \times 11 = 33 \).
In simple words: To check if a number is a perfect square, find its prime factors. If all factors come in pairs, it's a perfect square. So, 725 and 190 are not perfect squares, but 841 and 1089 are.

🎯 Exam Tip: Always perform prime factorization to determine if a number is a perfect square. A number is a perfect square only if every prime factor occurs an even number of times.

 

Question 5. Find the square root by prime factorisation method.
(i) 144
(ii) 256
(iii) 784
(iv) 1156
(v) 4761
(vi) 9025
Answer:
(i) First, we find the prime factors of 144:
\( 144 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \)
To find the square root, we take one factor from each pair:
\( \sqrt{144} = \sqrt{(2 \times 2) \times (2 \times 2) \times (3 \times 3)} = 2 \times 2 \times 3 = 12 \).
(ii) The prime factorization of 256 is:
\( 256 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \)
Grouping these factors into pairs, we take one from each pair:
\( \sqrt{256} = \sqrt{(2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (2 \times 2)} = 2 \times 2 \times 2 \times 2 = 16 \).
(iii) Let's find the prime factors of 784:
\( 784 = 2 \times 2 \times 2 \times 2 \times 7 \times 7 \)
For the square root, we pick one number from each pair of factors:
\( \sqrt{784} = \sqrt{(2 \times 2) \times (2 \times 2) \times (7 \times 7)} = 2 \times 2 \times 7 = 28 \).
(iv) The prime factorization of 1156 is:
\( 1156 = 2 \times 2 \times 17 \times 17 \)
We group these factors into pairs and take one number from each pair:
\( \sqrt{1156} = \sqrt{(2 \times 2) \times (17 \times 17)} = 2 \times 17 = 34 \).
(v) We break down 4761 into its prime factors:
\( 4761 = 3 \times 3 \times 23 \times 23 \)
To get the square root, we select one factor from each pair:
\( \sqrt{4761} = \sqrt{(3 \times 3) \times (23 \times 23)} = 3 \times 23 = 69 \).
(vi) The prime factorization of 9025 is:
\( 9025 = 5 \times 5 \times 19 \times 19 \)
We find the square root by choosing one number from each pair of identical factors:
\( \sqrt{9025} = \sqrt{(5 \times 5) \times (19 \times 19)} = 5 \times 19 = 95 \).
In simple words: For each number, break it into prime factors. Then, group the factors in pairs. The square root is found by multiplying one number from each pair. This gives: (i) 12, (ii) 16, (iii) 28, (iv) 34, (v) 69, (vi) 95.

🎯 Exam Tip: Always show the prime factorization steps clearly. Remember to take only one factor from each pair when finding the square root.

 

Question 6. Find the square root by long division method.
(i) 1764
(ii) 6889
(iii) 11025
(iv) 17956
(v) 418609
Answer:
(i) To find the square root of 1764 using the long division method:
Group the digits from the right in pairs: 17 64.
Find the largest square less than or equal to 17, which is 16 (\( 4^2 \)). Write 4 as the quotient.
Subtract 16 from 17 to get 1. Bring down the next pair, 64, making it 164.
Double the current quotient (4) to get 8. Find a digit (let's say 'x') such that \( 8x \times x \) is close to 164. Here, \( 82 \times 2 = 164 \). Write 2 next to 4 in the quotient.
Subtract 164 from 164 to get 0.
So, the square root of 1764 is 42.
(ii) To find the square root of 6889 using long division:
Group the digits in pairs from the right: 68 89.
The largest square less than or equal to 68 is 64 (\( 8^2 \)). Write 8 as the first digit of the quotient.
Subtract 64 from 68, leaving 4. Bring down the next pair, 89, to form 489.
Double the current quotient (8) to get 16. Now, find a digit 'x' such that \( 16x \times x \) is close to 489. Here, \( 163 \times 3 = 489 \). Write 3 as the next digit in the quotient.
Subtract 489 from 489, resulting in 0.
Therefore, the square root of 6889 is 83.
(iii) To find the square root of 11025 by long division:
Group the digits in pairs from the right: 1 10 25.
The largest square less than or equal to 1 is 1 (\( 1^2 \)). Write 1 as the first digit of the quotient.
Subtract 1 from 1, leaving 0. Bring down the next pair, 10.
Double the quotient (1) to get 2. Since 10 is too small for \( 2x \times x \), we write 0 as the next digit in the quotient and bring down the next pair, 25, making it 1025.
The doubled quotient is now 20. Find 'x' such that \( 20x \times x \) is close to 1025. Here, \( 205 \times 5 = 1025 \). Write 5 as the next digit in the quotient.
Subtract 1025 from 1025, resulting in 0.
So, the square root of 11025 is 105.
(iv) To find the square root of 17956 using the long division method:
Group the digits in pairs from the right: 1 79 56.
The largest square less than or equal to 1 is 1 (\( 1^2 \)). Write 1 as the first digit of the quotient.
Subtract 1 from 1, leaving 0. Bring down the next pair, 79.
Double the quotient (1) to get 2. Find a digit 'x' such that \( 2x \times x \) is close to 79. Here, \( 23 \times 3 = 69 \). Write 3 as the next digit in the quotient.
Subtract 69 from 79, leaving 10. Bring down the next pair, 56, making it 1056.
Double the current quotient (13) to get 26. Find 'x' such that \( 26x \times x \) is close to 1056. Here, \( 264 \times 4 = 1056 \). Write 4 as the next digit in the quotient.
Subtract 1056 from 1056, resulting in 0.
Therefore, the square root of 17956 is 134.
(v) To find the square root of 418609 using long division:
Group the digits in pairs from the right: 41 86 09.
The largest square less than or equal to 41 is 36 (\( 6^2 \)). Write 6 as the first digit of the quotient.
Subtract 36 from 41, leaving 5. Bring down the next pair, 86, to form 586.
Double the current quotient (6) to get 12. Find 'x' such that \( 12x \times x \) is close to 586. Here, \( 124 \times 4 = 496 \). Write 4 as the next digit in the quotient.
Subtract 496 from 586, leaving 90. Bring down the next pair, 09, to form 9009.
Double the current quotient (64) to get 128. Find 'x' such that \( 128x \times x \) is close to 9009. Here, \( 1287 \times 7 = 9009 \). Write 7 as the next digit in the quotient.
Subtract 9009 from 9009, resulting in 0.
Thus, the square root of 418609 is 647.
In simple words: Using the long division method, we find the square roots are: (i) 42, (ii) 83, (iii) 105, (iv) 134, (v) 647. This method involves grouping digits and systematically finding the root.

🎯 Exam Tip: Practice the long division method for square roots carefully. Remember to pair digits from the right, double the quotient at each step, and find the appropriate next digit.

 

Question 7. Estimate the value of the following square roots to the nearest whole number:
(i) \( \sqrt{440} \)
(ii) \( \sqrt{800} \)
(iii) \( \sqrt{1020} \)
Answer:
(i) To estimate \( \sqrt{440} \), we look for perfect squares close to 440. We know that \( 20^2 = 400 \) and \( 21^2 = 441 \). Since 440 is very close to 441, the square root of 440 is approximately 21. Estimating helps quickly find approximate values.
(ii) To estimate \( \sqrt{800} \), we find the perfect squares nearest to 800. We know that \( 28^2 = 784 \) and \( 29^2 = 841 \). Since 800 is closer to 784 than to 841, the square root of 800 is approximately 28. Knowing common squares speeds up estimation.
(iii) To estimate \( \sqrt{1020} \), we check perfect squares around 1020. We find that \( 31^2 = 961 \) and \( 32^2 = 1024 \). Since 1020 is much closer to 1024 than 961, the square root of 1020 is approximately 32. Estimation is useful for quick calculations.
In simple words: We find the nearest perfect squares to estimate. (i) \( \sqrt{440} \) is about 21 (close to 441). (ii) \( \sqrt{800} \) is about 28 (close to 784). (iii) \( \sqrt{1020} \) is about 32 (close to 1024).

🎯 Exam Tip: To estimate a square root, find the two consecutive perfect squares that the number lies between. The closest perfect square will give the best estimate for the square root.

 

Question 8. Find the square root of the following decimal numbers and fractions.
(i) 2.89
(ii) 67.24
(iii) 2.0164
(iv) \( \frac{144}{225} \)
(v) \( 7\frac{18}{49} \)
Answer:
(i) To find the square root of 2.89, we can use the long division method for decimals:
Pair the digits from the decimal point: 2. 89.
The largest square less than 2 is 1 (\( 1^2 \)). We write 1 as the first digit of the quotient. Subtract 1 from 2, leaving 1. Bring down 89, making it 189. Place a decimal point in the quotient.
Double 1 to get 2. We find that \( 27 \times 7 = 189 \). So, \( \sqrt{2.89} = 1.7 \). Understanding decimal placement is key.
(ii) To find \( \sqrt{67.24} \) by long division:
Group digits: 67. 24.
The largest square less than 67 is 64 (\( 8^2 \)). Write 8 in the quotient. Subtract 64 from 67, get 3. Bring down 24, making 324. Place a decimal point in the quotient.
Double 8 to get 16. Find 'x' such that \( 16x \times x \) is close to 324. Here, \( 162 \times 2 = 324 \). Write 2 in the quotient.
Subtract 324 from 324, get 0.
So, \( \sqrt{67.24} = 8.2 \). The method for decimals is similar to whole numbers.
(iii) To calculate \( \sqrt{2.0164} \) using long division:
Group digits: 2. 01 64.
The largest square less than 2 is 1 (\( 1^2 \)). Write 1 in the quotient. Subtract 1 from 2, get 1. Bring down 01, making 101. Place a decimal point.
Double 1 to get 2. Find 'x' such that \( 2x \times x \) is close to 101. Here, \( 24 \times 4 = 96 \). Write 4 in the quotient.
Subtract 96 from 101, get 5. Bring down 64, making 564.
Double 14 (ignoring decimal for doubling) to get 28. Find 'x' such that \( 28x \times x \) is close to 564. Here, \( 282 \times 2 = 564 \). Write 2 in the quotient.
Subtract 564 from 564, get 0.
So, \( \sqrt{2.0164} = 1.42 \). This method applies to numbers with many decimal places too.
(iv) To find the square root of a fraction like \( \frac{144}{225} \), we find the square root of the numerator and the denominator separately.
\( \sqrt{\frac{144}{225}} = \frac{\sqrt{144}}{\sqrt{225}} = \frac{12}{15} \). This property simplifies fraction square roots.
(v) First, convert the mixed fraction \( 7\frac{18}{49} \) into an improper fraction:
\( 7\frac{18}{49} = \frac{(7 \times 49) + 18}{49} = \frac{343 + 18}{49} = \frac{361}{49} \)
Now, find the square root of the numerator and the denominator separately:
\( \sqrt{\frac{361}{49}} = \frac{\sqrt{361}}{\sqrt{49}} = \frac{19}{7} \)
This can be written as the mixed fraction \( 2\frac{5}{7} \). Always convert mixed fractions before finding their square roots.
In simple words: For decimals, use long division, pairing digits from the decimal point. For fractions, take the square root of the top and bottom separately. For mixed fractions, convert to improper fractions first. The answers are (i) 1.7, (ii) 8.2, (iii) 1.42, (iv) \( \frac{12}{15} \), (v) \( 2\frac{5}{7} \).

🎯 Exam Tip: When working with square roots of decimals, always pair digits from the decimal point outwards. For fractions, remember that \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \).

 

Question 9. Find the least number that must be subtracted to 6666 so that it becomes a perfect square. Also, find the square root of the perfect square thus obtained.
Answer:
To find the least number to subtract from 6666 to make it a perfect square, we use the long division method to find its square root.
When we perform long division for 6666, we group the digits as 66 66.
First, we find the largest square less than or equal to 66, which is \( 8^2 = 64 \). We write 8 in the quotient.
Subtracting 64 from 66 leaves 2. We bring down the next pair, 66, making the number 266.
Double the current quotient (8) to get 16. We need to find a digit 'x' such that \( 16x \times x \) is less than or equal to 266. Here, \( 161 \times 1 = 161 \). We write 1 in the quotient.
Subtracting 161 from 266 gives a remainder of 105.
This remainder (105) is the least number that must be subtracted from 6666.
So, the perfect square number obtained is \( 6666 - 105 = 6561 \).
The square root of 6561 is 81. This method helps in finding both the remainder and the new perfect square.
In simple words: Using long division on 6666, we find a remainder of 105. Subtracting this 105 makes 6666 a perfect square, 6561. The square root of 6561 is 81.

🎯 Exam Tip: When asked to find the number to be subtracted, the remainder from the long division method is your answer. The square root of the new number is the quotient obtained.

 

Question 10. Find the least number by which 1800 should be multiplied so that it becomes a perfect square. Also, find the square root of the perfect square thus obtained.
Answer:
To find the least number to multiply 1800 by to make it a perfect square, we first find its prime factors.
The prime factorization of 1800 is \( 2 \times 2 \times 3 \times 3 \times 5 \times 5 \times 2 \).
When we group these factors into pairs, we see that one factor of 2 is left without a pair.
To make 1800 a perfect square, we need to multiply it by another 2 to complete the pair for the lonely 2.
So, the least number to multiply by is 2.
The new number obtained is \( 1800 \times 2 = 3600 \).
Now, to find the square root of 3600:
\( \sqrt{3600} = \sqrt{(2 \times 2) \times (3 \times 3) \times (5 \times 5) \times (2 \times 2)} \).
Taking one from each pair of factors: \( \sqrt{3600} = 2 \times 3 \times 5 \times 2 = 60 \).
This process is useful for transforming numbers into perfect squares.
In simple words: By prime factorization, 1800 has an unpaired '2'. So, we multiply by 2 to get 3600, which is a perfect square. The square root of 3600 is 60.

🎯 Exam Tip: For multiplication questions, use prime factorization. The "least number" to multiply by is the factor(s) needed to complete all pairs of prime factors.

Objective Type Questions

 

Question 11. The square of 43 ends with the digit .
(a) 9
(b) 6
(c) 4
(d) 3
Answer: (a) 9
In simple words: The square of 43 ends with the same digit as \( 3 \times 3 \), which is 9.

🎯 Exam Tip: To find the last digit of a square, only consider the last digit of the original number and square it. The last digit of that result is your answer.

 

Question 12. _____ is added to \( 24^2 \) to get \( 25^2 \).
(a) \( 4^2 \)
(b) \( 5^2 \)
(c) \( 6^2 \)
(d) \( 7^2 \)
Answer: (d) \( 7^2 \)
In simple words: We need to find the difference between \( 25^2 \) (which is 625) and \( 24^2 \) (which is 576). The difference is 49, which is \( 7^2 \).

🎯 Exam Tip: This question tests the identity \( (n+1)^2 - n^2 = 2n + 1 \). Here, \( 25^2 - 24^2 = 2(24) + 1 = 48 + 1 = 49 = 7^2 \).

 

Question 13. \( \sqrt{48} \) is approximately equal to .
(a) 5
(b) 6
(c) 7
Answer: (c) 7
In simple words: Since 48 is very close to 49, and \( \sqrt{49} = 7 \), then \( \sqrt{48} \) is approximately 7.

🎯 Exam Tip: To approximate a square root, find the nearest perfect square. The square root of that perfect square will be the closest whole number approximation.

 

Question 14. \( \sqrt{128}-\sqrt{98}+\sqrt{18} \)
(a) \( \sqrt{2} \)
(b) \( \sqrt{8} \)
(c) \( \sqrt{48} \)
(d) \( \sqrt{32} \)
Answer: (d) \( \sqrt{32} \)
In simple words: Break down each square root to terms with \( \sqrt{2} \). \( 8\sqrt{2} - 7\sqrt{2} + 3\sqrt{2} \) gives \( 4\sqrt{2} \). This is the same as \( \sqrt{32} \).

🎯 Exam Tip: Simplify square roots by factoring out perfect squares (e.g., \( \sqrt{128} = \sqrt{64 \times 2} = 8\sqrt{2} \)). Once simplified, combine the like terms.

 

Question 15. The number of digits in the square root of 123454321 is _____.
(a) 4
(b) 5
(c) 6
(d) 7
Answer: (b) 5
In simple words: The number 123454321 has 9 digits. When a number has an odd number of digits, its square root has \( \frac{\text{number of digits} + 1}{2} \) digits. So, \( \frac{9+1}{2} = 5 \) digits.

🎯 Exam Tip: For a number 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. Count the digits carefully.

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