Chapter 7, Problem 31SGR

To determine

To Simplify: The expression ${\left(256\right)}^{\frac{3}{4}}$

Answer to Problem 31SGR

  ${\left(256\right)}^{\frac{3}{4}}=64$

Explanation of Solution

Given information:

The expression ${\left(256\right)}^{\frac{3}{4}}$

Concept and Formula Used:

Rule of exponent.

  $\sqrt[n]{a}=a^{\frac{1}{n}},{\left(ab\right)}^n=a^n{b}^n$

BODMAS rule

Calculation:

The given expression is

  $\begin{array}{l}{\left(256\right)}^{\frac{3}{4}}={\left(2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\right)}^{\frac{3}{4}}\\ {\left(256\right)}^{\frac{3}{4}}={\left[{\left(2\right)}^8\right]}^{\frac{3}{4}}\\ \text{Using }{\left(a^m\right)}^n=a^{m\times n}\\ {\left(256\right)}^{\frac{3}{4}}={\left(2\right)}^{8\times \frac{3}{4}}\\ {\left(256\right)}^{\frac{3}{4}}={\left(2\right)}^{2\times 3}\\ {\left(256\right)}^{\frac{3}{4}}={\left(2\right)}^6\\ {\left(256\right)}^{\frac{3}{4}}=64\end{array}$

Conclusion:

  ${\left(256\right)}^{\frac{3}{4}}=64$