GSEB Class 8 Maths Solutions Chapter 12 ઘાત અને ઘાતાંક Exercise 12.1

Question 1. કિંમત શોધો: \(3^{-2}\)
Answer:
\( = \frac{1}{3^2} \)
\( = \frac{1}{3 \times 3} \)
\( = \frac{1}{9} \)
In simple words: To find the value of a negative exponent, you take 1 and divide it by the base number raised to the positive power. Then you calculate the simple multiplication.

Exam Tip: Remember that a negative exponent \(a^{-n}\) means \( \frac{1}{a^n} \), not a negative result. This is a common mistake to avoid.

Question 2. કિંમત શોધો: \( (-4)^{-2} \)
Answer:
\( = \frac{1}{(-4)^2} \)
\( = \frac{1}{(-4) \times (-4)} \)
\( = \frac{1}{16} \)
In simple words: When you have a negative exponent, flip the base to its reciprocal, making the exponent positive. Then, multiply the base by itself the number of times shown by the power.

Exam Tip: Pay close attention to negative bases. Squaring a negative number always gives a positive result.

Question 3. કિંમત શોધો: \( \left(\frac{1}{2}\right)^{-5} \)
Answer:
\( = \frac{1}{\left(\frac{1}{2}\right)^5} \)
\( = \frac{1}{\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2}} \)
\( = \frac{1}{\frac{1}{32}} \)
\( = 32 \)
In simple words: A negative exponent with a fraction means you can flip the fraction upside down and make the exponent positive. Then, raise the new fraction to that positive power.

Exam Tip: When dealing with fractional bases and negative exponents, you can invert the fraction and change the sign of the exponent to simplify the calculation.

Question 1. સાદું રૂપ આપો અને પરિણામને ધન ઘાતાંક સ્વરૂપે દર્શાવો: \( (-4)^5 \div (-4)^8 \)
Answer:
\( = (-4)^{5-8} \) (Using the rule: \( a^m \div a^n = a^{m-n} \))
\( = (-4)^{-3} \)
\( = \frac{1}{(-4)^3} \)
In simple words: When dividing numbers with the same base, you can subtract their powers. If the final power is negative, make it positive by putting 1 over the number raised to that positive power.

Exam Tip: Always remember the rules for exponents, especially for division, and ensure your final answer is expressed with a positive exponent as requested.

Question 2. સાદું રૂપ આપો અને પરિણામને ધન ઘાતાંક સ્વરૂપે દર્શાવો: \( \left(\frac{1}{2^3}\right)^2 \)
Answer:
\( = \frac{1^2}{(2^3)^2} \)
\( = \frac{1}{2^{3 \times 2}} \) (Using the rule: \( (a^m)^n = a^{mn} \))
\( = \frac{1}{2^6} \)
In simple words: When a power is raised to another power, you multiply the exponents. Also, if a fraction is raised to a power, both the top and bottom numbers get that power.

Exam Tip: A power to a power rule \((a^m)^n = a^{mn}\) is fundamental here. Ensure you apply it correctly to the exponent in the denominator.

Question 3. સાદું રૂપ આપો અને પરિણામને ધન ઘાતાંક સ્વરૂપે દર્શાવો: \( (-3)^4 \times \left(\frac{5}{3}\right)^4 \)
Answer:
\( = \left[(-3) \times \frac{5}{3}\right]^4 \) (Using the rule: \( a^m \times b^m = (ab)^m \))
\( = [(-1) \times 5]^4 \)
\( = (-1)^4 \times 5^4 \)
\( = 1 \times 5^4 \)
\( = 5^4 \)
In simple words: If two different bases have the same power, you can multiply the bases first and then apply the power. Remember that a negative number raised to an even power becomes positive.

Exam Tip: The rule \(a^m \times b^m = (ab)^m\) is very useful for simplifying expressions with different bases but the same exponent. Also, recall that any non-zero number raised to an even power is positive.

Question 4. સાદું રૂપ આપો અને પરિણામને ધન ઘાતાંક સ્વરૂપે દર્શાવો: \( (3^{-7} \div 3^{-10}) \times 3^{-5} \)
Answer:
\( = (3^{(-7)-(-10)}) \times 3^{-5} \)
\( = (3^{-7 + 10}) \times 3^{-5} \) (Using the rule: \( a^m \div a^n = a^{m-n} \))
\( = 3^3 \times 3^{-5} \)
\( = 3^{3+(-5)} \)
\( = 3^{-2} \) (Using the rule: \( a^m \times a^n = a^{m+n} \))
\( = \frac{1}{3^2} \)
In simple words: First, handle the division by subtracting the exponents. Then, for multiplication, add the exponents. If you end up with a negative exponent, rewrite it as a fraction with a positive exponent.

Exam Tip: Be careful with signs when subtracting negative exponents. A common error is writing \( -7 - 10 \) instead of \( -7 - (-10) \).

Question 5. સાદું રૂપ આપો અને પરિણામને ધન ઘાતાંક સ્વરૂપે દર્શાવો: \( 2^{-3} \times (-7)^{-3} \)
Answer:
\( = (2 \times (-7))^{-3} \) (Using the rule: \( a^m \times b^m = (ab)^m \))
\( = (-14)^{-3} \)
\( = \frac{1}{(-14)^3} \)
In simple words: When different numbers have the same negative exponent, you can multiply the numbers first and then apply the negative exponent. Afterwards, change the negative exponent into a positive one by making it a fraction.

Exam Tip: Remember to use the exponent rule \(a^m \times b^m = (ab)^m\) when the exponents are the same, even if they are negative. Always express the final answer with a positive exponent.

Question (i). કિંમત શોધો: \( (3^0 + 4^{-1}) \times 2^2 \)
Answer:
\( = (1 + \frac{1}{4}) \times 2^2 \)
\( = (\frac{4+1}{4}) \times 4 \)
\( = (\frac{5}{4}) \times 4 \)
\( = 5 \)
In simple words: Any number raised to the power of zero is 1. A negative exponent means to take the reciprocal. Then, follow the order of operations: simplify inside the parentheses first, and then multiply.

Exam Tip: Recall that any non-zero number raised to the power of 0 is 1. Also, convert negative exponents to positive ones (e.g., \(4^{-1} = \frac{1}{4}\)) before performing addition.

Question (ii). કિંમત શોધો: \( (2^{-1} \times 4^{-1}) \div 2^{-2} \)
Answer:
\( = (\frac{1}{2} \times \frac{1}{4}) \div \frac{1}{2^2} \)
\( = (\frac{1}{8}) \div \frac{1}{4} \)
\( = \frac{1}{8} \times \frac{4}{1} \)
\( = \frac{4}{8} \)
\( = \frac{1}{2} \)
In simple words: Convert all negative exponents to fractions with positive exponents. Multiply the fractions first, then change the division problem into multiplication by flipping the second fraction. Finally, simplify the result.

Exam Tip: When dividing by a fraction, multiply by its reciprocal. Ensure you correctly convert all negative exponents to their fractional forms at the start.

Question (iii). કિંમત શોધો: \( \left(\frac{1}{2}\right)^{-2} + \left(\frac{1}{3}\right)^{-2} + \left(\frac{1}{4}\right)^{-2} \)
Answer:
\( = \frac{1}{(\frac{1}{2})^2} + \frac{1}{(\frac{1}{3})^2} + \frac{1}{(\frac{1}{4})^2} \)
\( = \frac{1}{\frac{1}{4}} + \frac{1}{\frac{1}{9}} + \frac{1}{\frac{1}{16}} \)
\( = 4 + 9 + 16 \)
\( = 29 \)
In simple words: A negative exponent on a fraction means you can flip the fraction and make the exponent positive. Then, calculate each term and add them up.

Exam Tip: Remember that \( (\frac{a}{b})^{-n} = (\frac{b}{a})^n \). Apply this rule to quickly simplify each term before adding.

Question (iv). કિંમત શોધો: \( (3^{-1} + 4^{-1} + 5^{-1})^0 \)
Answer:
\( (3^{-1} + 4^{-1} + 5^{-1})^0 = 1 \)
In simple words: Any number (except zero) raised to the power of zero always equals 1. It doesn't matter what is inside the parentheses, as long as the whole thing is not zero.

Exam Tip: The rule \(a^0 = 1\) (where \(a \neq 0\)) is a quick way to solve such problems without needing to calculate the terms inside the parentheses.

Question (v). કિંમત શોધો: \( \left\{\left(\frac{-2}{3}\right)^{-2}\right\}^2 \)
Answer:
\( = \left(\frac{-2}{3}\right)^{(-2) \times 2} \) (Using the rule: \( (a^m)^n = a^{mn} \))
\( = \left(\frac{-2}{3}\right)^{-4} \)
\( = \left(\frac{3}{-2}\right)^4 \)
\( = \frac{3^4}{(-2)^4} \)
\( = \frac{3 \times 3 \times 3 \times 3}{(-2) \times (-2) \times (-2) \times (-2)} \)
\( = \frac{81}{16} \)
In simple words: First, multiply the exponents when a power is raised to another power. If the result is a negative exponent, flip the fraction inside and make the exponent positive. Then, calculate the new power.

Exam Tip: Remember that \( (\frac{a}{b})^{-n} = (\frac{b}{a})^n \). Also, an even power of a negative number will result in a positive number.

Question (i). કિંમત શોધો: \( \frac{8^{-1} \times 5^3}{2^{-4}} \), \( (t \neq 0) \)
Answer:
\( = \frac{(2^3)^{-1} \times 5^3}{2^{-4}} \)
\( = \frac{2^{-3} \times 5^3}{2^{-4}} \)
\( = 2^{-3-(-4)} \times 5^3 \)
\( = 2^{-3+4} \times 5^3 \)
\( = 2^1 \times 5^3 \)
\( = 2 \times 125 \)
\( = 250 \)
In simple words: Convert numbers like 8 into their prime factor form. Then, apply exponent rules for multiplication and division. Remember to subtract exponents when dividing with the same base, and convert negative exponents into positive ones by flipping them.

Exam Tip: Convert all numbers to their prime bases (e.g., \(8 = 2^3\)) to effectively apply exponent rules for simplification. Be careful with signs during exponent subtraction.

Question (ii). કિંમત શોધો: \( (5^{-1} \times 2^{-1}) \times 6^{-1} \)
Answer:
\( = (\frac{1}{5} \times \frac{1}{2}) \times \frac{1}{6} \)
\( = (\frac{1}{10}) \times \frac{1}{6} \)
\( = \frac{1}{60} \)
બીજી રીત :
\( = (5 \times 2)^{-1} \times 6^{-1} \)
\( = (10)^{-1} \times 6^{-1} \)
\( = (10 \times 6)^{-1} \)
\( = (60)^{-1} \)
\( = \frac{1}{60} \)
In simple words: You can either change all negative exponents into fractions first and then multiply them. Or, if the exponents are the same, multiply the bases and then apply the negative exponent, converting to a fraction at the end.

Exam Tip: The rule \(a^{-n} = \frac{1}{a^n}\) is key here. Also, applying \(a^m \times b^m = (ab)^m\) can offer an alternative, sometimes faster, path to the solution.

Question 5. જો \( 5^m \div 5^{-3} = 5^5 \) હોય, તો m શોધો.
Answer:
Given, \( 5^m \div 5^{-3} = 5^5 \)
We know that, \( a^p \div a^q = a^{p-q} \)
\( \implies 5^{m-(-3)} = 5^5 \)
\( \implies 5^{m+3} = 5^5 \)
Since the bases are the same, the exponents must be equal.
\( \implies m+3 = 5 \)
\( \implies m = 5-3 \)
\( \implies m = 2 \)
Thus, the value of m is 2.
In simple words: When you divide numbers with the same base, you subtract their powers. Since both sides of the equation have the same base (5), their powers must also be equal. Then, solve the simple equation to find 'm'.

Exam Tip: When the bases on both sides of an equation are equal, set the exponents equal to each other to solve for the unknown variable. Remember to handle negative exponents correctly during subtraction.

Question (i). કિંમત શોધો: \( \left\{\left(\frac{1}{3}\right)^{-1}-\left(\frac{1}{4}\right)^{-1}\right\}^{-1} \)
Answer:
\( = \left\{\frac{3}{1}-\frac{4}{1}\right\}^{-1} \) (Using the rule: \( a^{-m} = \frac{1}{a^m} \))
\( = \{3-4\}^{-1} \)
\( = \{-1\}^{-1} \)
\( = \frac{1}{-1} \)
\( = -1 \)
In simple words: First, deal with the negative exponents inside the brackets by flipping the fractions. Then, do the subtraction. Finally, apply the outermost negative exponent by taking the reciprocal of the result.

Exam Tip: Work from the innermost parentheses outwards. The rule \(a^{-1} = \frac{1}{a}\) is useful for simplifying terms with an exponent of -1.

Question (ii). કિંમત શોધો: \( \left(\frac{5}{8}\right)^{-7} \times \left(\frac{8}{5}\right)^{-4} \)
Answer:
\( = \left(\frac{5}{8}\right)^{-7} \times \left(\frac{5}{8}\right)^4 \) (Using the rule: \( a^{-m} = \frac{1}{a^m} \), and also \( (\frac{b}{a})^{-n} = (\frac{a}{b})^n \))
\( = \left(\frac{5}{8}\right)^{-7+4} \) (Using the rule: \( a^m \times a^n = a^{m+n} \))
\( = \left(\frac{5}{8}\right)^{-3} \)
\( = \left(\frac{8}{5}\right)^3 \)
\( = \frac{8^3}{5^3} \)
\( = \frac{8 \times 8 \times 8}{5 \times 5 \times 5} \)
\( = \frac{512}{125} \)
In simple words: Make sure both parts of the multiplication have the same base. You can flip a fraction with a negative exponent to make the exponent positive. Once the bases are the same, add the exponents, then calculate the final power.

Exam Tip: A key step is to make the bases the same. Remember that \( (\frac{b}{a})^{-n} = (\frac{a}{b})^n \) is a powerful way to achieve this, allowing you to then apply the product rule for exponents.

Question (i). સાદું રૂપ આપો: \( \frac{25 \times t^{-4}}{5^{-3} \times 10 \times t^{-8}} \), \( (t \neq 0) \)
Answer:
\( = \frac{5^2 \times t^{-4}}{5^{-3} \times (5 \times 2) \times t^{-8}} \)
\( = \frac{5^2 \times t^{-4}}{5^{-3+1} \times 2 \times t^{-8}} \)
\( = \frac{5^2 \times t^{-4}}{5^{-2} \times 2 \times t^{-8}} \)
\( = \frac{5^2}{5^{-2}} \times \frac{t^{-4}}{t^{-8}} \times \frac{1}{2} \)
\( = 5^{2-(-2)} \times t^{-4-(-8)} \times \frac{1}{2} \)
\( = 5^{2+2} \times t^{-4+8} \times \frac{1}{2} \)
\( = 5^4 \times t^4 \times \frac{1}{2} \)
\( = \frac{625 t^4}{2} \)
In simple words: Convert all numbers to their prime bases and use exponent rules to simplify. Group terms with the same base and apply rules for division (subtract exponents). Finally, combine the remaining terms.

Exam Tip: Break down composite numbers (like 25 and 10) into their prime factors before applying exponent rules. This helps in grouping like bases and simplifying the expression efficiently. Be careful with negative exponent subtraction.

Question (ii). સાદું રૂપ આપો: \( \frac{3^{-5} \times 10^{-5} \times 125}{5^{-7} \times 6^{-5}} \)
Answer:
\( = \frac{3^{-5} \times (2 \times 5)^{-5} \times 5^3}{5^{-7} \times (2 \times 3)^{-5}} \)
\( = \frac{3^{-5} \times 2^{-5} \times 5^{-5} \times 5^3}{5^{-7} \times 2^{-5} \times 3^{-5}} \)
\( = \frac{3^{-5} \times 2^{-5} \times 5^{-5+3}}{5^{-7} \times 2^{-5} \times 3^{-5}} \)
\( = \frac{3^{-5} \times 2^{-5} \times 5^{-2}}{5^{-7} \times 2^{-5} \times 3^{-5}} \)
\( = 3^{-5-(-5)} \times 2^{-5-(-5)} \times 5^{-2-(-7)} \)
\( = 3^0 \times 2^0 \times 5^{-2+7} \)
\( = 1 \times 1 \times 5^5 \)
\( = 5^5 \)
In simple words: First, convert all composite numbers (like 10, 125, and 6) into their prime factors. Then, use exponent rules to group terms with the same base. Simplify by subtracting exponents for division, and remember that any non-zero number raised to the power of zero is 1.

Exam Tip: Always decompose numbers into their prime factors (e.g., \(10 = 2 \times 5\), \(125 = 5^3\), \(6 = 2 \times 3\)) to effectively apply exponent laws. Be meticulous with the subtraction of negative exponents.