9.2 Simplify square roots  (Page 2/3)

Solution

$\begin{array}{cccc}& & & \sqrt{72n^7}\hfill \\ \begin{array}{c}\text{Rewrite the radicand as a product using the}\hfill \\ \text{largest perfect square factor.}\hfill \end{array}\hfill & & & \sqrt{36n^6·2n}\hfill \\ \begin{array}{c}\text{Rewrite the radical as the product of two}\hfill \\ \text{radicals.}\hfill \end{array}\hfill & & & \sqrt{36n^6}·\sqrt{2n}\hfill \\ \text{Simplify.}\hfill & & & 6n^3\sqrt{2n}\hfill \end{array}$

Solution

$\begin{array}{cccc}& & & \sqrt{63u^3{v}^5}\hfill \\ \begin{array}{c}\text{Rewrite the radicand as a product using the}\hfill \\ \text{largest perfect square factor.}\hfill \end{array}\hfill & & & \sqrt{9u^2{v}^4·7uv}\hfill \\ \begin{array}{c}\text{Rewrite the radical as the product of two}\hfill \\ \text{radicals.}\hfill \end{array}\hfill & & & \sqrt{9u^2{v}^4}·\sqrt{7uv}\hfill \\ \text{Simplify.}\hfill & & & 3u{v}^2\sqrt{7uv}\hfill \end{array}$

Solution

$\begin{array}{cccc}& & & 3+\sqrt{32}\hfill \\ \begin{array}{c}\text{Rewrite the radicand as a product using the}\hfill \\ \text{largest perfect square factor.}\hfill \end{array}\hfill & & & 3+\sqrt{16·2}\hfill \\ \begin{array}{c}\text{Rewrite the radical as the product of two}\hfill \\ \text{radicals.}\hfill \end{array}\hfill & & & 3+\sqrt{16}·\sqrt{2}\hfill \\ \text{Simplify.}\hfill & & & 3+4\sqrt{2}\hfill \end{array}$

The terms are not like and so we cannot add them. Trying to add an integer and a radical is like trying to add an integer and a variable—they are not like terms!

Solution

$\begin{array}{cccc}& & & \frac{4-\sqrt{48}}{2}\hfill \\ \\ \\ \begin{array}{c}\text{Rewrite the radicand as a product using the}\hfill \\ \text{largest perfect square factor.}\hfill \end{array}\hfill & & & \frac{4-\sqrt{16·3}}{2}\hfill \\ \\ \\ \begin{array}{c}\text{Rewrite the radical as the product of two}\hfill \\ \text{radicals.}\hfill \end{array}\hfill & & & \frac{4-\sqrt{16}·\sqrt{3}}{2}\hfill \\ \\ \\ \text{Simplify.}\hfill & & & \frac{4-4\sqrt{3}}{2}\hfill \\ \\ \\ \begin{array}{c}\text{Factor the common factor from the}\hfill \\ \text{numerator.}\hfill \end{array}\hfill & & & \frac{4\left(1-\sqrt{3}\right)}{2}\hfill \\ \\ \\ \begin{array}{c}\text{Remove the common factor, 2, from the}\hfill \\ \text{numerator and denominator.}\hfill \end{array}\hfill & & & \frac{\cancel{2}·2\left(1-\sqrt{3}\right)}{\cancel{2}}\hfill \\ \\ \\ \text{Simplify.}\hfill & & & 2\left(1-\sqrt{3}\right)\hfill \end{array}$

Solution

$\begin{array}{cccc}& & & \sqrt{\frac{9}{64}}\hfill \\ \text{Since}{\left(\frac{3}{8}\right)}^2=\frac{9}{64}\hfill & & & \frac{3}{8}\hfill \end{array}$

Solution

$\begin{array}{cccc}& & & \sqrt{\frac{45}{80}}\hfill \\ \begin{array}{c}\text{Simplify inside the radical first. Rewrite}\hfill \\ \text{showing the common factors of the}\hfill \\ \text{numerator and denominator.}\hfill \end{array}\hfill & & & \sqrt{\frac{5·9}{5·16}}\hfill \\ \begin{array}{c}\text{Simplify the fraction by removing common}\hfill \\ \text{factors.}\hfill \end{array}\hfill & & & \sqrt{\frac{9}{16}}\hfill \\ \text{Simplify.}{\left(\frac{3}{4}\right)}^2=\frac{9}{16}\hfill & & & \frac{3}{4}\hfill \end{array}$

Solution

$\begin{array}{cccc}& & & \sqrt{\frac{m^6}{m^4}}\hfill \\ \\ \\ \begin{array}{c}\text{Simplify the fraction inside the radical first.}\hfill \\ \\ \\ \text{Divide the like bases by subtracting the}\hfill \\ \text{exponents.}\hfill \end{array}\hfill & & & \sqrt{m^2}\hfill \\ \\ \\ \text{Simplify.}\hfill & & & m\hfill \end{array}$

Solution

$\begin{array}{cccc}& & & \sqrt{\frac{48p^7}{3p^3}}\hfill \\ \text{Simplify the fraction inside the radical first.}\hfill & & & \sqrt{16p^4}\hfill \\ \text{Simplify.}\hfill & & & 4p^2\hfill \end{array}$