Question

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

$$\begin{gathered} \left(a\right){\left(\frac{4}{9}\right)}^{-3} \\ \left(b\right){\left(-\frac{u}{v}\right)}^{-5} \end{gathered}$$

Short answer

The simplified expressions are:

$$\begin{gathered} \left(a\right){\left(\frac{4}{9}\right)}^{-3}=\frac{729}{64} \\ \left(b\right){\left(-\frac{u}{v}\right)}^{-5}=-\frac{v^5}{u^5} \end{gathered}$$

Step-by-step solution

Step 1. Property for negative exponent

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 ${\left(\frac{4}{9}\right)}^{-3}$.

We have to simplify the expression using property for negative exponents.

Part a Step 2. Simplify

Using property for negative exponents,

$\begin{array}{c}{\left(\frac{4}{9}\right)}^{-3}={\left(\frac{9}{4}\right)}^3\\ =\frac{729}{64}\end{array}$

So, the simplified expression is${\left(\frac{4}{9}\right)}^{-3}=\frac{729}{64}$

Part b Step 1. Given information

The given expression is ${\left(-\frac{u}{v}\right)}^{-5}$.

We have to simplify the expression using property for negative exponents.

Part b Step 2.Simplify

Using property for negative exponents,

$\begin{array}{c}{\left(-\frac{u}{v}\right)}^{-5}={\left(-\frac{v}{u}\right)}^5\\ =-\frac{v^5}{u^5}\end{array}$

So, the simplified expression is${\left(-\frac{u}{v}\right)}^{-5}=-\frac{v^5}{u^5}$