Chapter 7.4, Problem 64PPE

To determine

To simplify: the expression ${\left(\frac{9t^{\frac{2}{3}}}{36t}\right)}^3$

Answer to Problem 64PPE

The required answer is: $\frac{0.015625}{t}$

Explanation of Solution

Given information:

The given expression: ${\left(\frac{9t^{\frac{2}{3}}}{36t}\right)}^3$

Calculation:

  ${\left(\frac{9t^{\frac{2}{3}}}{36t}\right)}^3=\frac{{\left(9t^{\frac{2}{3}}\right)}^3}{{(36t)}^3}$

(Raise the numerator and denominator to the third power)

  $=\frac{9^3{t}^{\frac{2}{3}\cdot 3}}{{36}^3{t}^3}$

(Multiply the exponents in the numerator and denominator)

  $\begin{array}{l}=\frac{729t^2}{46656t^3}\\ =\frac{729}{46656}{t}^{2-3}\end{array}$

(Subtract exponents when dividing powers with the same base)

  $\begin{array}{l}=\frac{729}{46656}{t}^{-1}\\ =\frac{729}{46656}\cdot \frac{1}{t^1}\end{array}$

(Rewrite using positive exponents)

  $=\frac{0.015625}{t}$

(Simplify)