Chapter 7.3, Problem 76PPS

To determine

To simplify the given expression.

Answer to Problem 76PPS

  ${({81}^{\frac{1}{4}})}^{-2}=\frac{1}{9}$

Explanation of Solution

Given:

Expression: ${({81}^{\frac{1}{4}})}^{-2}$

Formula used:

  ${(a^b)}^c=a^{bc}$

Calculation:

Consider the given expression,

  $\Rightarrow {({81}^{\frac{1}{4}})}^{-2}$

Substituting the above formula in the given expression,

  $\begin{array}{l}$ \Rightarrow {({81}^{\frac{1}{4}})}^{-2}$\Rightarrow ({81}^{-\frac{2}{4}})\\ \Rightarrow ({81}^{-\frac{1}{2}})\\ $ \Rightarrow {(9^{\frac{1}{4}})}^{-2}$\Rightarrow 9^{\frac{-1\times 2}{2}}\\ \Rightarrow 9^{-1}\\ \Rightarrow \frac{1}{9}\end{array}$

Conclusion:

Therefore, the simplified form of the equation is $\frac{1}{9}$ .