Chapter 7.4, Problem 46PPE

To determine

To simplify: the expression ${\left(\frac{6n^2}{3n}\right)}^{-3}$

Answer to Problem 46PPE

The required answer is: $\frac{0.00058}{n^3}$

Explanation of Solution

Given information:

The given expression: ${\left(\frac{6n^2}{3n}\right)}^{-3}$

Calculation:

  ${\left(\frac{6n^2}{3n}\right)}^{-3}=\frac{1}{{\left(\frac{6n^2}{3n}\right)}^3}$

(Rewrite using positive exponents)

  $=\frac{1}{\frac{{(6n^2)}^3}{{(3n)}^3}}$

(Raise the quotient to third power)

  $=\frac{1}{\frac{{(6n)}^6}{{(3n)}^3}}$

(Multiply the exponents in the quotient)

  $=1\cdot \frac{{(3n)}^3}{{(6n)}^6}$

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

  $\begin{array}{l}=\frac{3^3{n}^3}{6^6{n}^6}\\ =\frac{3^3}{6^6}\times n^{3-6}\end{array}$

(Subtract exponents when dividing powers with the same base)

  $\begin{array}{l}=\frac{27}{46656}{n}^{-3}\\ =0.00058n^{-3}\\ =0.00058\times \frac{1}{n^3}\end{array}$

(Rewrite using positive exponents)

  $=\frac{0.00058}{n^3}$

(Simplify)