Question

Simplify Expressions Using the Properties for Exponents.

$$\begin{gathered} \left(a\right)r^{-3} \\ \left(b\right){10}^{-5} \\ \left(c\right)\frac{1}{q^{-10}} \\ \left(d\right)\frac{1}{{10}^{-3}} \end{gathered}$$

Short answer

The simplified form is:

$$\begin{gathered} \left(a\right)r^{-3}=\frac{1}{r^3} \\ \left(b\right){10}^{-5}=\frac{1}{100,000} \\ \left(c\right)\frac{1}{q^{-10}}=q^{10} \\ \left(d\right)\frac{1}{{10}^{-3}}=1,000 \end{gathered}$$

Step-by-step solution

Part (a) Step 1. Given that the negative exponent r-3 .

Properties of Negative Exponents : If $n$ is an integer and $a\ne 0$, then$a^{-n}=\frac{1}{a^n}or\frac{1}{a^n}=a^n$.

Step 2. We need to find the simplest form of given expression.

Use the definition of a negative exponent $a^{-n}=\frac{1}{a^n}$, to simplify the given expression.

$r^{-3}=\frac{1}{r^3}$.

Part (b) Step 1. Given that the negative exponent 10-5 .

Properties of Negative Exponents : If$n$is an integer and$a\ne 0$, then$a^{-n}=\frac{1}{a^n}or\frac{1}{a^n}=a^n$.

Step 2. We need to find the simplest form of given expression.

Use the definition of a negative exponent $a^{-n}=\frac{1}{a^n}$, to simplify the given expression.

$$\begin{gathered} {10}^{-5}=\frac{1}{{10}^5} \\ {10}^{-5}=\frac{1}{100,000} \end{gathered}$$

Part (c) Step 1. Given that the negative exponent 1q-10.

Properties of Negative Exponents : If$n$is an integer and $a\ne 0$, then$a^{-n}=\frac{1}{a^n}or\frac{1}{a^n}=a^n$.

Step 2. We need to find the simplest form of given expression.

Use the definition of a negative exponent $\frac{1}{a^{-n}}=a^n$, to simplify the given expression.

$\frac{1}{q^{-10}}=q^{10}$.

Part (d) Step 1. Given that the negative exponent 110-3.

Properties of Negative Exponents : If $n$is an integer and $a\ne 0$, then$a^{-n}=\frac{1}{a^n}or\frac{1}{a^n}=a^n$.

Step 2. We need to find the simplest form of given expression.

Use the definition of a negative exponent $\frac{1}{a^{-n}}=a^n$, to simplify the given expression.

$$\begin{gathered} \frac{1}{{10}^{-3}}={10}^3 \\ \frac{1}{{10}^{-3}}=1,000 \end{gathered}$$