Q. 105 - Intermediate Algebra

Question

Use the definition of  Negative Exponents to simplify (a) ${\left(-5\right)}^{-2}$ (b) $-5^{-2}$ (c) ${\left(-\frac{1}{5}\right)}^{-2}$ (d) $-{\left(\frac{1}{5}\right)}^{-2}$

Step-by-Step Solution

Answer

The simplified expression is (a) $\frac{1}{25}$ (b) $-\frac{1}{25}$ (c) $25$ (d) $-25$

Part (a). Step 1. Given information.

The given expression is ${\left(-5\right)}^{-2}$.

We have to simplify the equation.

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 exponent

$$\begin{gathered} {\left(-5\right)}^{-2}={\left(-\frac{1}{5}\right)}^2 \\ =\left(-\frac{1}{5}\right)·\left(-\frac{1}{5}\right) \\ =\frac{1}{25} \end{gathered}$$

So, ${\left(-5\right)}^{-2}=\frac{1}{25}.$

Part (b). Step 1. Given information.

The given expression is $-5^{-2}$.

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

$$\begin{gathered} -5^{-2}=-{\left(\frac{1}{5}\right)}^2 \\ =-\frac{1}{25} \end{gathered}$$

So, $-5^{-2}=-\frac{1}{25}$.

Part (c). Step 1. Given information.

The given expression is ${\left(-\frac{1}{5}\right)}^{-2}$.

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

$$\begin{gathered} {\left(-\frac{1}{5}\right)}^{-2}={\left(-5\right)}^2 \\ =\left(-5\right)·\left(-5\right) \\ =25 \end{gathered}$$

So, ${\left(-\frac{1}{5}\right)}^{-2}=25.$

Part (d). Step 1. Given information.

The given expression is $-{\left(\frac{1}{5}\right)}^{-2}$.

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

$$\begin{gathered} -{\left(\frac{1}{5}\right)}^{-2}=-{\left(5\right)}^2 \\ =-25 \end{gathered}$$

So, $-{\left(\frac{1}{5}\right)}^{-2}=-25.$