Chapter 7.4, Problem 42PPS

To determine

To simplify the expression $\sqrt{\frac{27}{m^5}}$ .

Answer to Problem 42PPS

  $\frac{3}{m^2}\cdot \sqrt{\frac{3}{m}}$

Explanation of Solution

Given:

  $\sqrt{\frac{27}{m^5}}$

Formula used:

The radical rules:

  $\sqrt[n]{a}=a^{\frac{1}{n}}$

  ${(a^m)}^n=a^{mn}$

  $\sqrt[a]{b\times c}=\sqrt[a]{b}\times \sqrt[a]{c}$

Calculation:

  $\begin{array}{l}\Rightarrow \sqrt{\frac{27}{m^5}}\\ \Rightarrow \sqrt[]{3\times 3\times 3\times \frac{1}{m\times m\times m\times m\times m}}\\ \Rightarrow \sqrt[]{3\times 3\times 3}\times \frac{1}{\sqrt{m\times m\times m\times m\times m}}[\because \sqrt[a]{b\times c}=\sqrt[a]{b}\times \sqrt[a]{c}]\end{array}$

  $\begin{array}{l}$ \Rightarrow {(3^3)}^{\frac{1}{2}}\times \frac{1}{{\left(m^5\right)}^{\frac{1}{2}}}[\because \sqrt[a]{b}={(b)}^{\frac{1}{a}}]$$ \Rightarrow {(3)}^{\frac{3}{2}}\times \frac{1}{{\left(m\right)}^{\frac{5}{2}}}[\because {(a^m)}^n=a^{mn}]$\Rightarrow 3\times \sqrt{3}\times \frac{1}{m^2\times \sqrt{m}}\\ \Rightarrow \frac{3}{m^2}\times \frac{\sqrt{3}}{\sqrt{m}}\\ \Rightarrow \frac{3}{m^2}\cdot \sqrt{\frac{3}{m}}\end{array}$

Conclusion:

The simplified form of the given expression is equal to $\frac{3}{m^2}\cdot \sqrt{\frac{3}{m}}$ .