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)${\left(-5\right)}^{-3}$

Short answer

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

Step-by-step solution

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