Question

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

$$\begin{gathered} \left(a\right)s^{-8} \\ \left(b\right){10}^{-2} \\ \left(c\right)\frac{1}{t^{-9}} \\ \left(d\right)\frac{1}{{10}^{-4}} \end{gathered}$$

Short answer

The simplified expressions are:

$$\begin{gathered} \left(a\right)s^{-8}=\frac{1}{s^8} \\ \left(b\right){10}^{-2}=\frac{1}{{10}^2} \\ \left(c\right)\frac{1}{t^{-9}}=t^9 \\ \left(d\right)\frac{1}{{10}^{-4}}={10}^4 \end{gathered}$$

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 $s^{-8}$.

We have to simplify the given expression using negative exponents

Part a Step 2. Simplify

Using Property for negative exponents,

$s^{-8}=\frac{1}{s^8}$

So, the simplified expression is$s^{-8}=\frac{1}{s^8}$

Part b Step 1. Given information

The given expression is${10}^{-2}$ .

We have to simplify the given expression using negative exponents

Part b Step 2. Simplify

Using Property for negative exponents,

${10}^{-2}=\frac{1}{{10}^2}$

So, the simplified expression is${10}^{-2}=\frac{1}{{10}^2}$

Part c Step 1. Given information 

The given expression is$\frac{1}{t^{-9}}$ .

We have to simplify the given expression using negative exponents

Part c Step 2. Simplify

Using Property for negative exponents,

$\frac{1}{t^{-9}}=t^9$

So, the simplified expression is$\frac{1}{t^{-9}}=t^9$

Part d Step 1. Given information

The given expression is $\frac{1}{{10}^{-4}}$ .

We have to simplify the given expression using negative exponents

Part d Step 2. Simplify

Using Property for negative exponents,

$\frac{1}{{10}^{-4}}={10}^4$

So, the simplified expression is$\frac{1}{{10}^{-4}}={10}^4$