Chapter 7.4, Problem 67PPS

To determine

To simplify the expression $\sqrt{\frac{1}{5}}-\sqrt{5}$ .

Answer to Problem 67PPS

  $\frac{-4}{\sqrt{5}}$

Explanation of Solution

Given:

  $\sqrt{\frac{1}{5}}-\sqrt{5}$

Formula used:

The radical rules:

  $\sqrt[n]{a}=a^{\frac{1}{n}}$

  ${(a^m)}^n=a^{mn}$

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

  $a^m\cdot a^n=a^{m+n}$

Calculation:

  $\begin{array}{l}\Rightarrow \sqrt{\frac{1}{5}}-\sqrt{5}\\ \Rightarrow \frac{\sqrt{1}}{\sqrt{5}}-\sqrt{5}[\because \sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}]\\ \Rightarrow \frac{\sqrt{1}-\sqrt{5}\times \sqrt{5}}{\sqrt{5}}\\ \Rightarrow \frac{\sqrt{1}-{(5)}^{\frac{1}{2}}\times {(5)}^{\frac{1}{2}}}{\sqrt{5}}[\because a^m\cdot a^n=a^{m+n}]\\ \Rightarrow \frac{1-{(5)}^{\frac{2}{2}}}{\sqrt{5}}\text{ [}\because \sqrt{1}\text{ = 1]}\\ \Rightarrow \frac{1-5^1}{\sqrt{5}}\\ \Rightarrow \frac{-4}{\sqrt{5}}\end{array}$

Conclusion:

The simplified form of the given expression is equal to $\frac{-4}{\sqrt{5}}.$