Chapter 7.5, Problem 24P

Write each expression in exponential form.

$\sqrt[4]{256b^5}$

To determine

To simplify the expression $\sqrt[4]{256b^5}$ in exponential form.

Answer to Problem 24P

  $\sqrt[4]{256b^5}=4b^{\frac{5}{4}}$

Explanation of Solution

Given information :

Given expression $\sqrt[4]{256b^5}$

Since $\sqrt[n]{a}=a^{\frac{1}{n}}\mathrm{............}(i),{\left(a^m\right)}^n=a^{m\times n}\mathrm{........}\text{(ii)}\text{and}{\left(ab\right)}^m=a^m{b}^m\mathrm{......}(iii)$

Now consider $\sqrt[4]{256b^5}={\left(256b^5\right)}^{\frac{1}{4}}\text{from}\text{rule}(i)$

  $\begin{array}{l}={256}^{\frac{1}{4}}{\left(b^5\right)}^{\frac{1}{4}}\text{from}\text{rule}\text{(iii)}\\ ={\left(4^4\right)}^{\frac{1}{4}}{\left(b^5\right)}^{\frac{1}{4}}\text{from}\text{rule}\text{(ii)}\\ ={\left(4\right)}^{4\times \frac{1}{4}}{\left(b\right)}^{5\times \frac{1}{4}}\\ \therefore \sqrt[4]{256b^5}=4b^{\frac{5}{4}}\end{array}$

Hence, $\sqrt[4]{256b^5}=4b^{\frac{5}{4}}$