Chapter 7.4, Problem 64PPS

To determine

To simplify the expression $\sqrt{\frac{1}{9}}-\sqrt{2}$ .

Answer to Problem 64PPS

  $\frac{1-3\sqrt{2}}{3}$

Explanation of Solution

Given:

  $\sqrt{\frac{1}{9}}-\sqrt{2}$

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{1}{9}}-\sqrt{2}\\ \Rightarrow \frac{1}{\sqrt{3\times 3}}-\sqrt{2}\\ $ \Rightarrow \frac{1}{{(3^2)}^{\frac{1}{2}}}-\sqrt{2}[\because \sqrt[a]{b}={(b)}^{\frac{1}{a}}]$\end{array}$

  $\begin{array}{l}$ \Rightarrow \frac{1}{{(3)}^{\frac{2}{2}}}-\sqrt{2}[\because {(a^m)}^n=a^{mn}]$\Rightarrow \frac{1}{{(3)}^{}}-\sqrt{2}\\ \Rightarrow \frac{1-3\sqrt{2}}{3}\end{array}$

Conclusion:

The simplified form of the given expression is equal to $\frac{1-3\sqrt{2}}{3}$ .