Chapter 3, Problem 1TE

In Exercises 1–6, use the rules of exponents to simplify the following expressions and remove all zero and negative exponents. Assume that all variables are nonzero.

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

To determine

To simplify: The expression ${\left(\frac{2}{3}\right)}^{-2}$ by using the rules of exponents and remove all zero and negative exponents.

Answer to Problem 1TE

The positive exponent form of the expression ${\left(\frac{2}{3}\right)}^{-2}$ is $\frac{9}{4}$.

Explanation of Solution

Property used:

Zero and negative exponents.

For $a\ne 0$ and $b\ne 0$,

1. $a^0=1$

2. $a^{-1}=\frac{1}{a}$

3. $a^{-n}=\frac{1}{a^n}$

4. ${\left(\frac{a}{b}\right)}^{-n}={\left(\frac{b}{a}\right)}^n$

Calculation:

The given expression is ${\left(\frac{2}{3}\right)}^{-2}$.

The expression ${\left(\frac{2}{3}\right)}^{-2}$ can be simplified by using Property 4 as follows.

$\begin{array}{c}{\left(\frac{2}{3}\right)}^{-2}={\left(\frac{3}{2}\right)}^2\\ =\frac{3^2}{2^2}\\ =\frac{9}{4}\end{array}$

Therefore, the positive exponent form of the expression ${\left(\frac{2}{3}\right)}^{-2}$ is $\frac{9}{4}$.