Chapter 7.4, Problem 59PPE

To determine

To write: property or properties of exponents to simplify the expression $\frac{{\left(2^{\frac{2}{3}}\right)}^3}{2^{15}}$

Answer to Problem 59PPE

The required answer is: 0.000122

Explanation of Solution

Given information:

The given expression: $\frac{{\left(2^{\frac{2}{3}}\right)}^3}{2^{15}}$

Calculation:

  $\frac{{\left(2^{\frac{2}{3}}\right)}^3}{2^{15}}=\frac{2^{\frac{2}{3}\cdot 3}}{2^{15}}$

(Multiply the exponents in the numerator)

  $\begin{array}{l}=\frac{2^2}{2^{15}}\\ =2^{2-15}\end{array}$

(Subtract exponents when dividing powers with the same base)

  $\begin{array}{l}=2^{-13}\\ =\frac{1}{2^{13}}\end{array}$

(Rewrite using positive exponents)

  $\begin{array}{l}=\frac{1}{8192}\\ =0.000122\end{array}$

(Simplify)