RBSE Solutions Class 8 Maths Chapter 3 Powers and Exponents Exercise 3.2

Question 1. Find the value of the following expressions.
(i) \( (5^{-1} \times 2^{-1}) \div 6^{-1} \)
(ii) \( \left(\frac{5}{6}\right)^6 \times \left(\frac{5}{6}\right)^{-4} \)
(iii) \( \left(\frac{8}{5}\right)^{-2} \times \left(\frac{8}{5}\right)^{-5} \)
(iv) \( \left(\frac{3}{5}\right)^{-2} \times \left(\frac{3}{5}\right)^{-3} \times \left(\frac{3}{5}\right)^0 \)
(v) \( \left(\frac{6}{15}\right)^3 \div \left(\frac{25}{32}\right)^2 \times \left(\frac{45}{16}\right) \)
Answer:
(i) To find the value of \( (5^{-1} \times 2^{-1}) \div 6^{-1} \):
First, change negative exponents to positive by taking the reciprocal.
\( (5^{-1} \times 2^{-1}) \div 6^{-1} \)
\( = \left(\frac{1}{5} \times \frac{1}{2}\right) \div \frac{1}{6} \)
Now, multiply inside the parentheses.
\( = \frac{1}{10} \div \frac{1}{6} \)
To divide by a fraction, multiply by its reciprocal.
\( = \frac{1}{10} \times \frac{6}{1} \)
\( = \frac{6}{10} \)
Simplify the fraction.
\( = \frac{3}{5} \)
The final answer is \( \frac{3}{5} \).
In simple words: Change negative powers to fractions. Multiply the first two fractions, then divide by the third one by flipping it and multiplying. Simplify the final fraction.

🎯 Exam Tip: Remember that \( a^{-n} = \frac{1}{a^n} \) is key to solving problems with negative exponents. Also, dividing by a fraction is the same as multiplying by its inverse.

Answer:
(ii) To find the value of \( \left(\frac{5}{6}\right)^6 \times \left(\frac{5}{6}\right)^{-4} \):
When multiplying powers with the same base, add the exponents.
\( \left(\frac{5}{6}\right)^6 \times \left(\frac{5}{6}\right)^{-4} \)
\( = \left(\frac{5}{6}\right)^{6 + (-4)} \)
\( = \left(\frac{5}{6}\right)^{6 - 4} \)
\( = \left(\frac{5}{6}\right)^2 \)
Now, calculate the square.
\( = \frac{5^2}{6^2} \)
\( = \frac{5 \times 5}{6 \times 6} \)
\( = \frac{25}{36} \)
The final answer is \( \frac{25}{36} \).
In simple words: When you multiply numbers with the same base but different powers, you add the powers together. Then, calculate the final power.

🎯 Exam Tip: The rule \( a^m \times a^n = a^{m+n} \) is fundamental for simplifying expressions with common bases. Pay attention to the signs of the exponents.

Answer:
(iii) To find the value of \( \left(\frac{8}{5}\right)^{-2} \times \left(\frac{8}{5}\right)^{-5} \):
When multiplying powers with the same base, add the exponents.
\( \left(\frac{8}{5}\right)^{-2} \times \left(\frac{8}{5}\right)^{-5} \)
\( = \left(\frac{8}{5}\right)^{-2 + (-5)} \)
\( = \left(\frac{8}{5}\right)^{-2 - 5} \)
\( = \left(\frac{8}{5}\right)^{-7} \)
Now, change the negative exponent to positive by taking the reciprocal of the base.
\( = \left(\frac{5}{8}\right)^7 \)
Calculate the power.
\( = \frac{5^7}{8^7} \)
\( = \frac{5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5}{8 \times 8 \times 8 \times 8 \times 8 \times 8 \times 8} \)
\( = \frac{78125}{2097152} \)
The final answer is \( \frac{78125}{2097152} \).
In simple words: Add the powers because the bases are the same. Then, flip the fraction to make the power positive and multiply the numbers out.

🎯 Exam Tip: Be careful with signs when adding negative exponents. Always convert negative exponents to positive ones by reciprocating the base before evaluating the power.

Answer:
(iv) To find the value of \( \left(\frac{3}{5}\right)^{-2} \times \left(\frac{3}{5}\right)^{-3} \times \left(\frac{3}{5}\right)^0 \):
When multiplying powers with the same base, add the exponents.
\( \left(\frac{3}{5}\right)^{-2} \times \left(\frac{3}{5}\right)^{-3} \times \left(\frac{3}{5}\right)^0 \)
\( = \left(\frac{3}{5}\right)^{-2 + (-3) + 0} \)
\( = \left(\frac{3}{5}\right)^{-2 - 3 + 0} \)
\( = \left(\frac{3}{5}\right)^{-5} \)
Now, change the negative exponent to positive by taking the reciprocal of the base.
\( = \left(\frac{5}{3}\right)^5 \)
Calculate the power.
\( = \frac{5^5}{3^5} \)
\( = \frac{5 \times 5 \times 5 \times 5 \times 5}{3 \times 3 \times 3 \times 3 \times 3} \)
\( = \frac{3125}{243} \)
The final answer is \( \frac{3125}{243} \).
In simple words: Add all the powers together because the numbers being multiplied are the same. Remember that any number raised to the power of zero is one. Then flip the fraction to get a positive power and solve.

🎯 Exam Tip: Any non-zero number raised to the power of zero is always 1. This rule, \( a^0 = 1 \), simplifies calculations significantly.

Answer:
(v) To find the value of \( \left(\frac{6}{15}\right)^3 \div \left(\frac{25}{32}\right)^2 \times \left(\frac{45}{16}\right) \):
Simplify the first fraction \( \frac{6}{15} \) by dividing by 3: \( \frac{2}{5} \).
Then express all numbers as powers of their prime factors.
\( = \left(\frac{2}{5}\right)^3 \div \left(\frac{5^2}{2^5}\right)^2 \times \left(\frac{3^2 \times 5}{2^4}\right) \)
Apply the power rules.
\( = \frac{2^3}{5^3} \div \frac{(5^2)^2}{(2^5)^2} \times \frac{3^2 \times 5}{2^4} \)
\( = \frac{2^3}{5^3} \div \frac{5^4}{2^{10}} \times \frac{3^2 \times 5}{2^4} \)
To divide, multiply by the reciprocal.
\( = \frac{2^3}{5^3} \times \frac{2^{10}}{5^4} \times \frac{3^2 \times 5}{2^4} \)
Combine terms with the same base by adding/subtracting exponents.
\( = \frac{2^{3+10} \times 3^2 \times 5^1}{5^3 \times 5^4 \times 2^4} \)
\( = \frac{2^{13} \times 3^2 \times 5^1}{5^{3+4} \times 2^4} \)
\( = \frac{2^{13} \times 3^2 \times 5^1}{5^7 \times 2^4} \)
Now, subtract exponents for division.
\( = 2^{13-4} \times 3^2 \times 5^{1-7} \)
\( = 2^9 \times 3^2 \times 5^{-6} \)
\( = 512 \times 9 \times \frac{1}{5^6} \)
\( = \frac{4608}{15625} \)
The final answer is \( \frac{4608}{15625} \).
In simple words: First, simplify any fractions. Then, use rules of exponents to combine terms with the same base by adding or subtracting powers. Finally, calculate the numerical value.

🎯 Exam Tip: Always simplify fractions and express numbers as prime factors first. This makes applying exponent rules much easier and reduces the chance of errors in large calculations.

Question 2. Simplify the following expressions.
(i) \( \frac{16^{-1} \times 5^3}{2^{-4}} \)
(ii) \( \frac{25xt^4}{5^3 \times 5xt^{-8}} \), where \( t \ne 0 \)
(iii) \( \frac{6^3 \times 7^4 \times 8^5}{4^5 \times 9^2 \times 16} \)
(iv) \( \frac{15^3 \times 18^2}{3^5 \times 5^4 \times 12^2} \)
Answer:
(i) To simplify \( \frac{16^{-1} \times 5^3}{2^{-4}} \):
First, express all numbers as powers of their prime factors and change negative exponents to positive ones.
\( = \frac{(2^4)^{-1} \times 5^3}{2^{-4}} \)
\( = \frac{2^{-4} \times 5^3}{2^{-4}} \)
Now, use the rule \( a^m \div a^n = a^{m-n} \).
\( = 2^{-4 - (-4)} \times 5^3 \)
\( = 2^{-4 + 4} \times 5^3 \)
\( = 2^0 \times 5^3 \)
Remember \( 2^0 = 1 \).
\( = 1 \times 5 \times 5 \times 5 \)
\( = 125 \)
The simplified value is 125.
In simple words: Convert everything to powers of 2 and 5. Flip numbers with negative powers. Then combine like terms by subtracting powers. Any number to the power of zero is one.

🎯 Exam Tip: Always convert larger bases (like 16) into prime factor bases (like 2) when dealing with exponents. This helps in applying exponent rules efficiently.

Answer:
(ii) To simplify \( \frac{25xt^4}{5^3 \times 5xt^{-8}} \):
First, express 25 as \( 5^2 \) and combine terms in the denominator.
\( = \frac{5^2 \times x^1 \times t^4}{5^3 \times 5^1 \times x^1 \times t^{-8}} \)
Combine the powers of 5 and x in the denominator.
\( = \frac{5^2 \times x^1 \times t^4}{5^{3+1} \times x^1 \times t^{-8}} \)
\( = \frac{5^2 \times x^1 \times t^4}{5^4 \times x^1 \times t^{-8}} \)
Now, subtract the exponents for terms with the same base.
\( = 5^{2-4} \times x^{1-1} \times t^{4-(-8)} \)
\( = 5^{-2} \times x^0 \times t^{4+8} \)
\( = 5^{-2} \times 1 \times t^{12} \)
Change the negative exponent to positive.
\( = \frac{1}{5^2} \times t^{12} \)
\( = \frac{t^{12}}{25} \)
The simplified expression is \( \frac{t^{12}}{25} \).
In simple words: Turn all numbers into powers of 5. For variables, combine them by subtracting their powers. Remember that any variable to the power of zero is one, and a negative power means it goes to the bottom of a fraction.

🎯 Exam Tip: Pay close attention to the signs of exponents when subtracting. Also, remember that \( x^0 = 1 \) for any non-zero \( x \).

Answer:
(iii) To simplify \( \frac{6^3 \times 7^4 \times 8^5}{4^5 \times 9^2 \times 16} \):
Express all bases as powers of prime factors (2, 3, 7).
\( = \frac{(2 \times 3)^3 \times 7^4 \times (2^3)^5}{(2^2)^5 \times (3^2)^2 \times 2^4} \)
Apply the power of a power rule \( (a^m)^n = a^{mn} \).
\( = \frac{2^3 \times 3^3 \times 7^4 \times 2^{15}}{2^{10} \times 3^4 \times 2^4} \)
Combine terms with the same base by adding exponents.
\( = \frac{2^{3+15} \times 3^3 \times 7^4}{2^{10+4} \times 3^4} \)
\( = \frac{2^{18} \times 3^3 \times 7^4}{2^{14} \times 3^4} \)
Now, subtract exponents for terms with the same base.
\( = 2^{18-14} \times 3^{3-4} \times 7^4 \)
\( = 2^4 \times 3^{-1} \times 7^4 \)
Change the negative exponent to positive.
\( = 16 \times \frac{1}{3} \times 2401 \)
\( = \frac{16 \times 2401}{3} \)
\( = \frac{38416}{3} \)
The simplified expression is \( \frac{38416}{3} \).
In simple words: Break down all numbers into their smallest prime parts (like 2, 3, 7). Add or subtract the powers of the same base numbers. If you get a negative power, move it to the bottom of the fraction. Finally, multiply the remaining numbers.

🎯 Exam Tip: This type of problem requires careful application of multiple exponent rules: prime factorization, power of a product, power of a power, and combining powers with the same base.

Answer:
(iv) To simplify \( \frac{15^3 \times 18^2}{3^5 \times 5^4 \times 12^2} \):
Express all bases as powers of prime factors (2, 3, 5).
\( = \frac{(3 \times 5)^3 \times (2 \times 3^2)^2}{3^5 \times 5^4 \times (2^2 \times 3)^2} \)
Apply the power of a product and power of a power rules.
\( = \frac{3^3 \times 5^3 \times 2^2 \times (3^2)^2}{3^5 \times 5^4 \times (2^2)^2 \times 3^2} \)
\( = \frac{3^3 \times 5^3 \times 2^2 \times 3^4}{3^5 \times 5^4 \times 2^4 \times 3^2} \)
Combine terms with the same base by adding exponents.
\( = \frac{2^2 \times 3^{3+4} \times 5^3}{2^4 \times 3^{5+2} \times 5^4} \)
\( = \frac{2^2 \times 3^7 \times 5^3}{2^4 \times 3^7 \times 5^4} \)
Now, subtract exponents for terms with the same base.
\( = 2^{2-4} \times 3^{7-7} \times 5^{3-4} \)
\( = 2^{-2} \times 3^0 \times 5^{-1} \)
Remember \( 3^0 = 1 \) and change negative exponents to positive.
\( = \frac{1}{2^2} \times 1 \times \frac{1}{5^1} \)
\( = \frac{1}{4} \times \frac{1}{5} \)
\( = \frac{1}{20} \)
The simplified expression is \( \frac{1}{20} \).
In simple words: Break down all numbers into their prime factors. Use the rules for powers to combine numbers with the same base by adding or subtracting their powers. Any number to the power of zero is one, and negative powers mean the number goes to the bottom of a fraction.

🎯 Exam Tip: Prime factorization of each base is crucial. For example, \( 12 = 2^2 \times 3 \). This step helps in systematically applying exponent rules for simplification.

Question 3. Find the value of x.
(i) \( \left(\frac{4}{3}\right)^{-4} \times \left(\frac{4}{3}\right)^{-5} = \left(\frac{4}{3}\right)^{-3x} \)
(ii) \( 7^x \div 7^{-3} = 7^5 \)
(iii) \( (4)^{2x+1} \div 16 = 64 \)
Answer:
(i) Given the equation: \( \left(\frac{4}{3}\right)^{-4} \times \left(\frac{4}{3}\right)^{-5} = \left(\frac{4}{3}\right)^{-3x} \)
When multiplying powers with the same base, add the exponents.
\( \left(\frac{4}{3}\right)^{-4 + (-5)} = \left(\frac{4}{3}\right)^{-3x} \)
\( \left(\frac{4}{3}\right)^{-9} = \left(\frac{4}{3}\right)^{-3x} \)
Since the bases are equal, their exponents must also be equal.
\( -9 = -3x \)
To find x, divide both sides by -3.
\( x = \frac{-9}{-3} \)
\( x = 3 \)
The value of x is 3.
In simple words: Since the bases (the big numbers) are the same on both sides, the powers (the small numbers) must also be equal. Add the powers on the left side, then set that total equal to the power on the right side and solve for x.

🎯 Exam Tip: The principle of equating exponents when bases are the same (if \( a^m = a^n \), then \( m=n \)) is fundamental for solving exponential equations.

Answer:
(ii) Given the equation: \( 7^x \div 7^{-3} = 7^5 \)
When dividing powers with the same base, subtract the exponents.
\( 7^{x - (-3)} = 7^5 \)
\( 7^{x+3} = 7^5 \)
Since the bases are equal, their exponents must also be equal.
\( x+3 = 5 \)
Subtract 3 from both sides to find x.
\( x = 5 - 3 \)
\( x = 2 \)
The value of x is 2.
In simple words: When you divide numbers with the same base, you subtract their powers. Do this on the left side, then set the resulting power equal to the power on the right side and find x.

🎯 Exam Tip: Always be careful with the signs when subtracting exponents, especially when a negative exponent is involved, e.g., \( x - (-3) = x+3 \).

Answer:
(iii) Given the equation: \( (4)^{2x+1} \div 16 = 64 \)
First, express all numbers as powers of the same base. Here, the base is 4.
\( 16 = 4^2 \)
\( 64 = 4^3 \)
Substitute these into the equation:
\( (4)^{2x+1} \div 4^2 = 4^3 \)
When dividing powers with the same base, subtract the exponents.
\( 4^{(2x+1) - 2} = 4^3 \)
\( 4^{2x - 1} = 4^3 \)
Since the bases are equal, their exponents must also be equal.
\( 2x - 1 = 3 \)
Add 1 to both sides.
\( 2x = 3 + 1 \)
\( 2x = 4 \)
Divide by 2 to find x.
\( x = \frac{4}{2} \)
\( x = 2 \)
The value of x is 2.
In simple words: Change all the big numbers (16 and 64) into powers of 4. Then, use the rule that when you divide numbers with the same base, you subtract their powers. After that, set the powers on both sides of the equal sign to be the same and solve for x.

🎯 Exam Tip: When solving exponential equations, the first step is often to rewrite all terms with a common base. This allows you to equate the exponents and solve for the variable.

Question 4. Find the value.
(i) \( \frac{3125 \times 1296}{6561 \times 1875} \)
(ii) \( \frac{1536 \times 972}{486 \times 1152} \)
Answer:
(i) To find the value of \( \frac{3125 \times 1296}{6561 \times 1875} \):
First, express each number as a power of its prime factors.
\( 3125 = 5^5 \)
\( 1296 = 2^4 \times 3^4 \)
\( 6561 = 3^8 \)
\( 1875 = 3 \times 5^4 \)
Substitute these into the expression:
\( = \frac{5^5 \times (2^4 \times 3^4)}{3^8 \times (3 \times 5^4)} \)
\( = \frac{5^5 \times 2^4 \times 3^4}{3^8 \times 3^1 \times 5^4} \)
Combine powers with the same base in the denominator.
\( = \frac{5^5 \times 2^4 \times 3^4}{3^{8+1} \times 5^4} \)
\( = \frac{5^5 \times 2^4 \times 3^4}{3^9 \times 5^4} \)
Now, subtract the exponents for terms with the same base.
\( = 2^4 \times 3^{4-9} \times 5^{5-4} \)
\( = 2^4 \times 3^{-5} \times 5^1 \)
Change the negative exponent to positive.
\( = 16 \times \frac{1}{3^5} \times 5 \)
\( = 16 \times \frac{1}{243} \times 5 \)
\( = \frac{16 \times 5}{243} \)
\( = \frac{80}{243} \)
The value of the expression is \( \frac{80}{243} \).
In simple words: Break each number into its prime factors. Use the rules of powers to add or subtract the exponents of the same base numbers. If you get a negative power, move that base to the bottom of the fraction. Finally, multiply the remaining numbers.

🎯 Exam Tip: Factorizing each number into its prime components (e.g., \( 1296 = 2^4 \times 3^4 \)) is a critical first step for these types of simplification problems.

Answer:
(ii) To find the value of \( \frac{1536 \times 972}{486 \times 1152} \):
First, express each number as a product of its prime factors.
\( 1536 = 2^9 \times 3 \)
\( 972 = 2^2 \times 3^5 \)
\( 486 = 2 \times 3^5 \)
\( 1152 = 2^7 \times 3^2 \)
Substitute these into the expression:
\( = \frac{(2^9 \times 3) \times (2^2 \times 3^5)}{(2 \times 3^5) \times (2^7 \times 3^2)} \)
Combine powers with the same base in the numerator and denominator by adding exponents.
\( = \frac{2^{9+2} \times 3^{1+5}}{2^{1+7} \times 3^{5+2}} \)
\( = \frac{2^{11} \times 3^6}{2^8 \times 3^7} \)
Now, subtract the exponents for terms with the same base.
\( = 2^{11-8} \times 3^{6-7} \)
\( = 2^3 \times 3^{-1} \)
Change the negative exponent to positive.
\( = 8 \times \frac{1}{3} \)
\( = \frac{8}{3} \)
The value of the expression is \( \frac{8}{3} \).
In simple words: Break down each number into its prime factors. Combine the powers of the same prime factors by adding their exponents. Then, subtract the powers of the same factors between the top and bottom of the fraction. A negative power means the factor goes to the bottom.

🎯 Exam Tip: Using prime factorization to simplify large numbers is a powerful technique. Double-check your factorization steps to avoid errors early in the calculation.