Question

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

$$\begin{gathered} \left(a\right){\left(\frac{3}{10}\right)}^{-2} \\ \left(b\right){\left(-\frac{2}{z}\right)}^{-3} \end{gathered}$$

Short answer

The simplified expressions are:

$$\begin{gathered} \left(a\right){\left(\frac{3}{10}\right)}^{-2}=\frac{100}{9} \\ \left(b\right){\left(-\frac{2}{z}\right)}^{-3}=-\frac{z^3}{8} \end{gathered}$$

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 information

The given expression is ${\left(\frac{3}{10}\right)}^{-2}$.

We have to simplify the expression using negative exponents.

Part a Step 2. Simplify

Using the property for negative exponents,

$\begin{array}{c}{\left(\frac{3}{10}\right)}^{-2}={\left(\frac{10}{3}\right)}^2\\ =\frac{100}{9}\end{array}$

So, the simplified expression is${\left(\frac{3}{10}\right)}^{-2}=\frac{100}{9}$

Part b Step 1. Given information

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

We have to simplify the expression using negative exponents.

Part b Step 2. Simplify

Using the property for negative exponents,

$\begin{array}{c}{\left(-\frac{2}{z}\right)}^{-3}={\left(-\frac{z}{2}\right)}^3\\ =-\frac{z^3}{8}\end{array}$

So, the simplified expression is${\left(-\frac{2}{z}\right)}^{-3}=-\frac{z^3}{8}$