Chapter 7.4, Problem 69PPE

To determine

To simplify: the expression ${\left(\frac{3}{x^2}\right)}^{-3}$ in three different ways

Answer to Problem 69PPE

The required answer is: $\frac{x^6}{27}$

Explanation of Solution

Given information:

The given expression: ${\left(\frac{3}{x^2}\right)}^{-3}$

Calculation:

First way:

  ${\left(\frac{3}{x^2}\right)}^{-3}=\frac{1}{{\left(\frac{3}{x^2}\right)}^3}$

(Rewrite using positive exponents)

  $=\frac{1}{\frac{{(3)}^3}{{(x^2)}^3}}$

(Raise the quotient to third power)

  $=\frac{1}{\frac{3^3}{x^6}}$

(Multiply the exponents)

  $=1\cdot \frac{x^6}{3^3}$

(Multiply by the reciprocal $\frac{3^3}{x^6}$ of which is $\frac{x^6}{3^3}$ )

  $=\frac{x^6}{27}$

(Simplify)

Second way:

  ${\left(\frac{3}{x^2}\right)}^{-3}=\frac{{(3)}^{-3}}{{(x^2)}^{-3}}$

(Raise the quotient to third power)

  $=\frac{3^{-3}}{x^{-6}}$

(Multiply the exponents in the denominator)

  $=\frac{1}{3^3}\cdot \frac{x^6}{1}$

(Rewrite using positive exponents)

  $\begin{array}{l}=\frac{x^6}{3^3}\\ =\frac{x^6}{27}\end{array}$

(Simplify)

Third way:

  ${\left(\frac{3}{x^2}\right)}^{-3}=\frac{{(3)}^{-3}}{{(x^2)}^{-3}}$

(Raise the quotient to third power)

  $=\frac{1}{{(3)}^3}\cdot \frac{{(x^2)}^3}{1}$

(Rewrite using positive exponents)

  $\begin{array}{l}=\frac{{(x^2)}^3}{{(3)}^3}\\ =\frac{x^6}{3^3}\end{array}$

(Multiply the exponents in the numerator)

  $=\frac{x^6}{27}$

(Simplify)