Question

Use the definition of a Negative Exponent to simplify$\left(a\right){\left(\frac{7}{2}\right)}^{-3}\left(b\right){\left(-\frac{3}{x}\right)}^{-3}$.

Short answer

The simplified equation is$\left(a\right)\frac{8}{343}\left(b\right)-\frac{x^3}{27}.$

Step-by-step solution

Part a Step 1. Given information

The given expression is ${\left(\frac{7}{2}\right)}^{-3}$.

We have to simplify the expression.

Part a Step 2. Definition of Negative Exponent.

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

$$\begin{gathered} {\left(\frac{7}{2}\right)}^{-3}={\left(\frac{2}{7}\right)}^3 \\ =\frac{8}{343} \end{gathered}$$

So,${\left(\frac{7}{2}\right)}^{-3}=\frac{8}{343}.$

Part b. Step 1. Given information.

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

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 exponent

$$\begin{gathered} {\left(-\frac{3}{x}\right)}^{-3}={\left(-\frac{x}{3}\right)}^3 \\ =-\frac{x^3}{27}. \end{gathered}$$

So,${\left(-\frac{3}{x}\right)}^{-3}=-\frac{x^3}{27}.$