Chapter 7.4, Problem 37PPS

To determine

To simplify the expression $\sqrt{75q{r}^3}$ .

Answer to Problem 37PPS

  $5\sqrt{3q{r}^3}$

Explanation of Solution

Given:

  $\sqrt{75q{r}^3}$

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{75q{r}^3}\\ \Rightarrow \sqrt[]{3\times 5\times 5\times q\times r\times r\times r}\\ \Rightarrow \sqrt[]{3}\times \sqrt{5\times 5}\times \sqrt[]{q}\times \sqrt{r\times r\times r}[\because \sqrt[a]{b\times c}=\sqrt[a]{b}\times \sqrt[a]{c}]\\ $ \Rightarrow {(3)}^{\frac{1}{2}}\times {(5^2)}^{\frac{1}{2}}\times {(q)}^{\frac{1}{2}}\times {(r^3)}^{\frac{1}{2}}[\because \sqrt[a]{b}={(b)}^{\frac{1}{a}}]$$ \Rightarrow {(3)}^{\frac{1}{2}}\times {(5)}^{\frac{2}{2}}\times {(q)}^{\frac{1}{2}}\times {(r)}^{\frac{3}{2}}[\because {(a^m)}^n=a^{mn}]$$ \Rightarrow {(3)}^{\frac{1}{2}}\times (5)\times {(q)}^{\frac{1}{2}}\times {(r)}^{\frac{3}{2}}$\Rightarrow 5\sqrt{3q{r}^3}\end{array}$

Conclusion:

The simplified form of the given expression is equal to $5\sqrt{3q{r}^3}$ .