QuestionJuly 15, 2025

Simplify. Write your answers without exponents. ((1)/(81))^-(3)/(4)= square 3(2)/(3)-(2)/(5)= square

Simplify. Write your answers without exponents. ((1)/(81))^-(3)/(4)= square 3(2)/(3)-(2)/(5)= square
Simplify. Write your answers without exponents. ((1)/(81))^-(3)/(4)= square 3(2)/(3)-(2)/(5)= square

Solution
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Answer

27, \frac{1}{4} Explanation 1. Simplify the first expression Use the property a^{-b} = \frac{1}{a^b} to rewrite (\frac{1}{81})^{-\frac{3}{4}} as 81^{\frac{3}{4}}. Then, find the fourth root of 81, which is 3, and raise it to the power of 3: 3^3 = 27. 2. Simplify the second expression Use the property a^{-b} = \frac{1}{a^b} to rewrite 32^{-\frac{2}{5}} as \frac{1}{32^{\frac{2}{5}}}. Find the fifth root of 32, which is 2, and square it: 2^2 = 4. Thus, \frac{1}{4}.

Explanation

1. Simplify the first expression<br /> Use the property $a^{-b} = \frac{1}{a^b}$ to rewrite $(\frac{1}{81})^{-\frac{3}{4}}$ as $81^{\frac{3}{4}}$. Then, find the fourth root of 81, which is 3, and raise it to the power of 3: $3^3 = 27$.<br /><br />2. Simplify the second expression<br /> Use the property $a^{-b} = \frac{1}{a^b}$ to rewrite $32^{-\frac{2}{5}}$ as $\frac{1}{32^{\frac{2}{5}}}$. Find the fifth root of 32, which is 2, and square it: $2^2 = 4$. Thus, $\frac{1}{4}$.
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