Chapter 6.2, Problem 16OE

To determine

To calculate: The simplified form of the expression ${\left(-\frac{4}{n^2}\right)}^3$.

Answer to Problem 16OE

The simplified form of the expression ${\left(-\frac{4}{n^2}\right)}^3$ is $-\frac{64}{n^6}$.

Explanation of Solution

Given information:

The expression ${\left(-\frac{4}{n^2}\right)}^3$.

Formula used:

Law of exponent for a quotient raised to power m where m is a positive integer is ${\left(\frac{a}{b}\right)}^m=\frac{a^m}{b^m}$.

Law of exponent for a power raised to power m where m is a positive integer is ${\left(a^n\right)}^m=a^{nm}$.

Calculation:

Consider the expression ${\left(-\frac{4}{n^2}\right)}^3$.

Recall that law of exponent for a quotient raised to power m where m is a positive integer is ${\left(\frac{a}{b}\right)}^m=\frac{a^m}{b^m}$.

Apply it,

  $\begin{array}{c}{\left(-\frac{4}{n^2}\right)}^3={\left(-1\right)}^3\frac{{\left(4\right)}^3}{{\left(n^2\right)}^3}\\ =-\frac{{\left(4\right)}^3}{{\left(n^2\right)}^3}\end{array}$

Recall that law of exponent for a power raised to power m where m is a positive integer is ${\left(a^n\right)}^m=a^{nm}$.

Apply it,

  $\begin{array}{c}{\left(-\frac{4}{n^2}\right)}^3={\left(-1\right)}^3\frac{{\left(4\right)}^3}{{\left(n^2\right)}^3}\\ =-\frac{{\left(4\right)}^3}{{\left(n^2\right)}^3}\\ =-\frac{64}{n^6}\end{array}$

Thus, the simplified form of the expression ${\left(-\frac{4}{n^2}\right)}^3$ is $-\frac{64}{n^6}$.