Question

Use the Definition of a Negative Exponent In the following exercises, simplify each expression.

$$\begin{gathered} \left(a\right){\left(\frac{5}{8}\right)}^{-2} \\ \left(b\right){\left(-\frac{b}{a}\right)}^{-2} \end{gathered}$$

Short answer

The simplified expressions are:

$$\begin{gathered} \left(a\right){\left(\frac{5}{8}\right)}^{-2}=\frac{64}{25} \\ \left(b\right){\left(-\frac{b}{a}\right)}^{-2}=\frac{a^2}{b^2} \end{gathered}$$

Step-by-step solution

Step 1. Property for negative exponents

If $n$is an integer and $a\ne 0$, then

$a^{-n}=\frac{1}{a^n}or\frac{1}{a^{-n}}=a^n$

Part a Step 1. Given expression 

The given expression is ${\left(\frac{5}{8}\right)}^{-2}$

We have to simplify the given expression using negative exponents.

Part a Step 2. Simplify

Using a Property for negative exponents,

$\begin{array}{c}{\left(\frac{5}{8}\right)}^{-2}={\left(\frac{8}{5}\right)}^2\\ =\frac{64}{25}\end{array}$

So, the simplified expression is${\left(\frac{5}{8}\right)}^{-2}=\frac{64}{25}$

Part b Step 1. Given information

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

We have to simplify the given expression using negative exponents.

Part b Step 2. Simplify

Using a Property for negative exponents,

$\begin{array}{c}{\left(-\frac{b}{a}\right)}^{-2}={\left(-\frac{a}{b}\right)}^2\\ =\frac{a^2}{b^2}\end{array}$

So, the simplified expression is${\left(-\frac{b}{a}\right)}^{-2}=\frac{a^2}{b^2}$