Chapter A.10, Problem 69AYU

To determine

To find: Simplify the expression ${\left(\frac{8}{9}\right)}^{-\frac{3}{2}}$.

Answer to Problem 69AYU

The simplest form of ${\left(\frac{8}{9}\right)}^{-\frac{3}{2}}$ is $\frac{27}{16\sqrt{2}}$.

Explanation of Solution

Given:

It is asked to simplify the expression ${\left(\frac{8}{9}\right)}^{-\frac{3}{2}}$.

Calculation:

If $a$ is a real number and $m$ and $n$ are integers containing no common factors, with $n\ge 2$, then $a^{\frac{m}{n}}=\sqrt[n]{a^m}={\left(\sqrt[n]{a}\right)}^m$ provided that $\sqrt[n]{a}$ exists.

Consider ${\left(\frac{8}{9}\right)}^{-\frac{3}{2}}$.

${\left(\frac{8}{9}\right)}^{-\frac{3}{2}}=\frac{{\left(8\right)}^{-\frac{3}{2}}}{{\left(9\right)}^{-\frac{3}{2}}}=\frac{{\left(9\right)}^{\frac{3}{2}}}{{\left(8\right)}^{\frac{3}{2}}}=\frac{{\left(\sqrt{9}\right)}^3}{{\left(\sqrt{8}\right)}^3}=\frac{3^2}{{\left(2\sqrt{2}\right)}^3}=\frac{27}{16\sqrt{2}}$