Chapter 7.4, Problem 43PPS

To determine

To simplify the expression $\sqrt{\frac{32}{t^4}}$ .

Answer to Problem 43PPS

  $\frac{4\sqrt{2}}{t^2}$

Explanation of Solution

Given:

  $\sqrt{\frac{32}{t^4}}$

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:

  $\begin{array}{l}\Rightarrow \sqrt{\frac{32}{t^4}}\\ \Rightarrow \sqrt[]{2\times 2\times 2\times 2\times 2\times \frac{1}{t\times t\times t\times t}}\\ \Rightarrow \sqrt[]{2\times 2\times 2\times 2\times 2}\times \frac{1}{\sqrt{t\times t\times t\times t}}[\because \sqrt[a]{b\times c}=\sqrt[a]{b}\times \sqrt[a]{c}]\end{array}$

  $\begin{array}{l}$ \Rightarrow {(2^5)}^{\frac{1}{2}}\times \frac{1}{{\left(t^4\right)}^{\frac{1}{2}}}[\because \sqrt[a]{b}={(b)}^{\frac{1}{a}}]$$ \Rightarrow {(2)}^{\frac{5}{2}}\times \frac{1}{{\left(t\right)}^{\frac{4}{2}}}[\because {(a^m)}^n=a^{mn}]$\Rightarrow 2^2\times \sqrt{2}\times \frac{1}{t^2}\\ \Rightarrow \frac{4\sqrt{2}}{t^2}\end{array}$

Conclusion:

The simplified form of the given expression is equal to $\frac{4\sqrt{2}}{t^2}$ .