Q. 106 - Intermediate Algebra

Question

Use the definition of Negative Exponents to simplify (a) $-5^{-3}$ (b) ${\left(-\frac{1}{5}\right)}^{-3}$ (c)$-{\left(\frac{1}{5}\right)}^{-3}$ (d) ${\left(-5\right)}^{-3}$

Step-by-Step Solution

Answer

The simplified expression is (a) $-\frac{1}{125}$ (b) $-125$ (c) $-125$ (d) $-\frac{1}{125}$.

Part (a). Step 1. Given Information.

The given expression is $-5^{-3}$.

We have to simplify the expression.

Part (a). Step 2. Definition of Negative Exponents.

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

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

Part (a). Step 3. Simplify the expression.

Using the definition of Negative Exponents

$$\begin{gathered} -5^{-3}=-{\left(\frac{1}{5}\right)}^{-3} \\ =-\left(\frac{1}{5}·\frac{1}{5}·\frac{1}{5}\right) \\ =-\frac{1}{125} \end{gathered}$$

So, $-5^{-3}=-\frac{1}{125}.$

Part (b). Step 1. Given information.

The given expression is ${\left(-\frac{1}{5}\right)}^{-3}$.

We have to simplify the expression.

Part (b). Step 2. Definition of Negative Exponents.

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

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

Part (b). Step 3. Simplify the expression.

Using the definition of Negative Exponents

$$\begin{gathered} {\left(-\frac{1}{5}\right)}^{-3}={\left(-5\right)}^3 \\ =\left(-5\right)·\left(-5\right)·\left(-5\right) \\ =-125 \end{gathered}$$

So, ${\left(-\frac{1}{5}\right)}^{-3}=-125.$

Part (c). Step 1. Given information.

The given expression is $-{\left(\frac{1}{5}\right)}^{-3}.$

We have to simplify the expression.

Part (c). Step 2. Definition of Negative Exponents.

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

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

Part (c). Step 3. Simplify the expression.

Using the definition of Negative Exponents

$$\begin{gathered} -{\left(\frac{1}{5}\right)}^{-3}=-{\left(5\right)}^3 \\ =-\left(5\right)·\left(5\right)·\left(5\right) \\ =-125 \end{gathered}$$

So, $-{\left(\frac{1}{5}\right)}^{-3}=-125.$

Part (d). Step 1. Given information.

The given expression is ${\left(-5\right)}^{-3}.$

We have to simplify the expression.

Part (d). Step 2. Definition of Negative Exponents.

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

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

Part (d). Step 3. Simplify the expression.

Using the definition of Negative Exponents

$$\begin{gathered} {\left(-5\right)}^{-3}={\left(-\frac{1}{5}\right)}^3 \\ =\left(-\frac{1}{5}\right)·\left(-\frac{1}{5}\right)·\left(-\frac{1}{5}\right) \\ =-\frac{1}{125} \end{gathered}$$

So, ${\left(-5\right)}^{-3}=-\frac{1}{125}.$