Q. 104 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

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

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