Chapter 7.4, Problem 83PFA

To determine

To simplify the expression $\sqrt{\frac{40}{12}}$ .

Answer to Problem 83PFA

  $\sqrt{\frac{10}{3}}$

Explanation of Solution

Given:

  $\sqrt{\frac{40}{12}}$

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{40}{12}}\\ \Rightarrow \frac{\sqrt{40}}{\sqrt{12}}\\ \Rightarrow \frac{\sqrt{2\times 2\times 2\times 5}}{\sqrt{2\times 2\times 3}}[\because \sqrt[a]{b\times c}=\sqrt[a]{b}\times \sqrt[a]{c}]\\ \Rightarrow \frac{\sqrt{2^2}\times \sqrt{2}\times \sqrt{5}}{\sqrt{2^2}\times \sqrt{3}}\\ $ \Rightarrow \frac{{(2^2)}^{\frac{1}{2}}\times {(2)}^{\frac{1}{2}}\times {(5)}^{\frac{1}{2}}}{{(2^2)}^{\frac{1}{2}}\times {(3)}^{\frac{1}{2}}}[\because \sqrt[a]{b}={(b)}^{\frac{1}{a}}]$$ \Rightarrow \frac{2^{\frac{2}{2}}\times {(2)}^{\frac{1}{2}}\times {(5)}^{\frac{1}{2}}}{2^{\frac{2}{2}}\times {(3)}^{\frac{1}{2}}}[\because {(a^m)}^n=a^{mn}]$\Rightarrow \frac{{(2)}^{\frac{1}{2}}\times {(5)}^{\frac{1}{2}}}{{(3)}^{\frac{1}{2}}}\\ \Rightarrow \frac{\sqrt{2}\times \sqrt{5}}{\sqrt{3}}\\ \Rightarrow \frac{\sqrt{10}}{\sqrt{3}}\\ \Rightarrow \sqrt{\frac{10}{3}}\end{array}$

Conclusion:

The simplified form of the given expression is equal to $\sqrt{\frac{10}{3}}$ .