Chapter 12.5, Problem 9MRE

To determine

Write the expression ${\left(x^{\frac{2}{3}}\right)}^{-\frac{3}{4}}$ in simplest form without negative exponents.

Answer to Problem 9MRE

The simplest form of expression is $\frac{x^{\frac{1}{2}}}{x}$ .

Explanation of Solution

Given information: The given expression is,

  ${\left(x^{\frac{2}{3}}\right)}^{-\frac{3}{4}}\mathrm{....}\left(1\right)$

Calculation: Simplify (1),

  $\begin{array}{c}{\left(x^{\frac{2}{3}}\right)}^{-\frac{3}{4}}={\left(x\right)}^{\frac{2}{3}\times -\frac{3}{4}}\\ ={\left(x\right)}^{-\frac{1}{2}}\\ =\left(\frac{1}{x^{\frac{1}{2}}}\right)\left(\frac{x^{\frac{1}{2}}}{x^{\frac{1}{2}}}\right)\mathrm{....}\left(\text{Rationalized}\right)\\ =\frac{x^{\frac{1}{2}}}{x}\end{array}$

Hence the simplest form of expression is $\frac{x^{\frac{1}{2}}}{x}$ .