8.2 Simplify Radical Expressions

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$\begin{array}{cccc}& & & \hfill \sqrt{500}\hfill \\ \begin{array}{c}\text{Rewrite the radicand as a product}\hfill \\ \text{using the largest perfect square factor.}\hfill \end{array}\hfill & & & \hfill \sqrt{100·5}\hfill \\ \begin{array}{c}\text{Rewrite the radical as the product of two}\hfill \\ \text{radicals}\hfill \end{array}\hfill & & & \hfill \sqrt{100}·\sqrt{5}\hfill \\ \text{Simplify.}\hfill & & & \hfill 10\sqrt{5}\hfill \end{array}$

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$\begin{array}{cccc}& & & \hfill \sqrt[3]{16}\hfill \\ \begin{array}{c}\text{Rewrite the radicand as a product using}\hfill \\ \text{the greatest perfect cube factor.}{2}^3=8\hfill \end{array}\hfill & & & \hfill \sqrt[3]{8·2}\hfill \\ \begin{array}{c}\text{Rewrite the radical as the product of two}\hfill \\ \text{radicals.}\hfill \end{array}\hfill & & & \hfill \sqrt[3]{8}·\sqrt[3]{2^{}}\hfill \\ \text{Simplify.}\hfill & & & \hfill 2\sqrt[3]{2^{}}\hfill \end{array}$

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$\begin{array}{cccc}& & & \hfill \sqrt[4]{243}\hfill \\ \begin{array}{c}\text{Rewrite the radicand as a product using}\hfill \\ \text{the greatest perfect fourth power factor.}\hfill \end{array}\hfill & & & \hfill \sqrt[4]{81·3^{}}\hfill \\ 3^4=81\hfill & & & \\ \begin{array}{c}\text{Rewrite the radical as the product of two}\hfill \\ \text{radicals}\hfill \end{array}\hfill & & & \hfill \sqrt[4]{81}·\sqrt[4]{3^{}}\hfill \\ \text{Simplify.}\hfill & & & \hfill 3\sqrt[4]{3^{}}\hfill \end{array}$

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$\begin{array}{cccc}& & & \hfill \sqrt{x^3}\hfill \\ \begin{array}{c}\text{Rewrite the radicand as a product using}\hfill \\ \text{the largest perfect square factor.}\hfill \end{array}\hfill & & & \hfill \sqrt{x^2·x}\hfill \\ \begin{array}{c}\text{Rewrite the radical as the product of two}\hfill \\ \text{radicals.}\hfill \end{array}\hfill & & & \hfill \sqrt{x^2}·\sqrt{x}\hfill \\ \text{Simplify.}\hfill & & & \hfill \left|x\right|\sqrt{x}\hfill \end{array}$

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$\begin{array}{cccc}& & & \hfill \sqrt[3]{x^4}\hfill \\ \begin{array}{c}\text{Rewrite the radicand as a product}\hfill \\ \text{using the largest perfect cube factor.}\hfill \end{array}\hfill & & & \hfill \sqrt[3]{x^3·x}.\hfill \\ \begin{array}{c}\text{Rewrite the radical as the product of two}\hfill \\ \text{radicals.}\hfill \end{array}\hfill & & & \hfill \sqrt[3]{x^3}·\sqrt[3]{x}\hfill \\ \text{Simplify.}\hfill & & & \hfill x\sqrt[3]{x}\hfill \end{array}$

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$\begin{array}{cccc}& & & \hfill \sqrt[4]{x^7}\hfill \\ \begin{array}{c}\text{Rewrite the radicand as a product}\hfill \\ \text{using the greatest perfect fourth power}\hfill \end{array}\hfill & & & \hfill \sqrt[4]{x^4·x^3}\hfill \\ \text{factor.}\hfill & & & \\ \begin{array}{c}\text{Rewrite the radical as the product of two}\hfill \\ \text{radicals.}\hfill \end{array}\hfill & & & \hfill \sqrt[4]{x^4}·\sqrt[4]{x^3}\hfill \\ \text{Simplify.}\hfill & & & \hfill \left|x\right|\sqrt[4]{x^3}\hfill \end{array}$

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

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$\begin{array}{cccc}& & & \hfill \sqrt[3]{24x^7}\hfill \\ \begin{array}{c}\text{Rewrite the radicand as a product}\hfill \\ \text{using perfect cube factors.}\hfill \end{array}\hfill & & & \hfill \sqrt[3]{8x^6·3x}\hfill \\ \begin{array}{c}\text{Rewrite the radical as the product of two}\hfill \\ \text{radicals.}\hfill \end{array}\hfill & & & \hfill \sqrt[3]{8x^6}·\sqrt[3]{3^{}x}\hfill \\ \text{Rewrite the first radicand as}{\left(2x^2\right)}^3.\hfill & & & \hfill \sqrt[3]{{\left(2^{}{x}^2\right)}^3}·\sqrt[3]{3^{}x}\hfill \\ \text{Simplify.}\hfill & & & \hfill 2x^2\sqrt[3]{3^{}x}\hfill \end{array}$

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$\begin{array}{cccc}& & & \hfill \sqrt[4]{80y^{14}}\hfill \\ \begin{array}{c}\text{Rewrite the radicand as a product}\hfill \\ \text{using perfect fourth power factors.}\hfill \end{array}\hfill & & & \hfill \sqrt[4]{16y^{12}·5^{}{y}^2}\hfill \\ \begin{array}{c}\text{Rewrite the radical as the product of two}\hfill \\ \text{radicals.}\hfill \end{array}\hfill & & & \hfill \sqrt[4]{16y^{12}}·\sqrt[4]{5y^2{}^{}}\hfill \\ \text{Rewrite the first radicand as}{\left(2y^3\right)}^4.\hfill & & & \hfill \sqrt[4]{{\left(2^{}{y}^3\right)}^4}·\sqrt[4]{5y^2{}^{}}\hfill \\ \text{Simplify.}\hfill & & & \hfill 2\left|y^3\right|\sqrt[4]{5^{}{y}^2}\hfill \end{array}$

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

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$\begin{array}{cccc}& & & \hfill \sqrt[3]{40x^4{y}^5}\hfill \\ \begin{array}{c}\text{Rewrite the radicand as a product}\hfill \\ \text{using the largest perfect cube factor.}\hfill \end{array}\hfill & & & \hfill \sqrt[3]{8x^3{y}^3·5x{y}^2}\hfill \\ \begin{array}{c}\text{Rewrite the radical as the product of two}\hfill \\ \text{radicals.}\hfill \end{array}\hfill & & & \hfill \sqrt[3]{8x^3{y}^3}·\sqrt[3]{5x{y}^2}\hfill \\ \text{Rewrite the first radicand as}{\left(2xy\right)}^3.\hfill & & & \hfill \sqrt[3]{{\left(2xy\right)}^3}·\sqrt[3]{5x{y}^2}\hfill \\ \text{Simplify.}\hfill & & & \hfill 2xy\sqrt[3]{5x{y}^2}\hfill \end{array}$

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$\begin{array}{cccc}& & & \hfill \sqrt[4]{48x^4{y}^7}\hfill \\ \begin{array}{c}\text{Rewrite the radicand as a product}\hfill \\ \text{using the largest perfect fourth power}\hfill \end{array}\hfill & & & \hfill \sqrt[4]{16x^4{y}^4·3y^3}\hfill \\ \text{factor.}\hfill & & & \\ \begin{array}{c}\text{Rewrite the radical as the product of two}\hfill \\ \text{radicals.}\hfill \end{array}\hfill & & & \hfill \sqrt[4]{16x^4{y}^4}·\sqrt[4]{3y^3}\hfill \\ \text{Rewrite the first radicand as}{\left(2xy\right)}^4.\hfill & & & \hfill \sqrt[4]{{\left(2xy\right)}^4}·\sqrt[4]{3y^3}\hfill \\ \text{Simplify.}\hfill & & & \hfill 2\left|xy\right|\sqrt[4]{3y^3}\hfill \end{array}$

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$\begin{array}{cccc}& & & \hfill \sqrt[3]{\mathrm{-27}}\hfill \\ \begin{array}{c}\text{Rewrite the radicand as a product using}\hfill \\ \text{perfect cube factors.}\hfill \end{array}\hfill & & & \hfill \sqrt[3]{{\left(\mathrm{-3}\right)}^3}\hfill \\ \text{Take the cube root.}\hfill & & & \hfill \mathrm{-3}\hfill \end{array}$

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$\begin{array}{cccc}& & & \hfill \sqrt[4]{\mathrm{-16}}\hfill \\ \text{There is no real number}n\text{where}{n}^4=\mathrm{-16}.\hfill & & & \hfill \text{Not a real number.}\hfill \end{array}$

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

The terms cannot be added as one has a radical and the other does not. Trying to add an integer and a radical is like trying to add an integer and a variable. They are not like terms!

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

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

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$\begin{array}{cccc}& & & \hfill \sqrt[3]{\frac{16}{54}}\hfill \\ \begin{array}{c}\text{Simplify inside the radical first.}\hfill \\ \text{Rewrite showing the common factors of}\hfill \\ \text{the numerator and denominator.}\hfill \end{array}\hfill & & & \hfill \sqrt[3]{\frac{2·8}{2·27}}\hfill \\ \begin{array}{c}\text{Simplify the fraction by removing}\hfill \\ \text{common factors.}\hfill \end{array}\hfill & & & \hfill \sqrt[3]{\frac{8}{27}}\hfill \\ \text{Simplify. Note}{\left(\frac{2}{3}\right)}^3=\frac{8}{27}.\hfill & & & \hfill \frac{2}{3}\hfill \end{array}$

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$\begin{array}{cccc}& & & \hfill \sqrt[4]{\frac{5}{80}}\hfill \\ \begin{array}{c}\text{Simplify inside the radical first.}\hfill \\ \text{Rewrite showing the common factors of}\hfill \\ \text{the numerator and denominator.}\hfill \end{array}\hfill & & & \hfill \sqrt[4]{\frac{5·1}{5·16}}\hfill \\ \begin{array}{c}\text{Simplify the fraction by removing}\hfill \\ \text{common factors.}\hfill \end{array}\hfill & & & \hfill \sqrt[4]{\frac{1}{16}}\hfill \\ \text{Simplify. Note}{\left(\frac{1}{2}\right)}^4=\frac{1}{16}.\hfill & & & \hfill \frac{1}{2}\hfill \end{array}$

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

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$\begin{array}{cccc}& & & \hfill \sqrt[3]{\frac{a^8}{a^5}}\hfill \\ \\ \\ \begin{array}{c}\text{Use the Quotient Property of exponents to}\hfill \\ \text{simplify the fraction under the radical first.}\hfill \end{array}\hfill & & & \hfill \sqrt[3]{a^3}\hfill \\ \\ \\ \text{Simplify.}\hfill & & & \hfill a\hfill \end{array}$

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$\begin{array}{cccc}& & & \hfill \sqrt[4]{\frac{a^{10}}{a^2}}\hfill \\ \\ \\ \begin{array}{c}\text{Use the Quotient Property of exponents to}\hfill \\ \text{simplify the fraction under the radical first.}\hfill \end{array}\hfill & & & \hfill \sqrt[4]{a^8}\hfill \\ \\ \\ \begin{array}{c}\text{Rewrite the radicand using perfect}\hfill \\ \text{fourth power factors.}\hfill \end{array}\hfill & & & \hfill \sqrt[4]{{\left(a^2\right)}^4}\hfill \\ \\ \\ \text{Simplify.}\hfill & & & \hfill a^2\hfill \end{array}$

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$\begin{array}{cccc}& & & \hfill \sqrt{\frac{45x^5}{y^4}}\hfill \\ \begin{array}{c}\text{We cannot simplify the fraction in the}\hfill \\ \text{radicand. Rewrite using the Quotient}\hfill \\ \text{Property.}\hfill \end{array}\hfill & & & \hfill \frac{\sqrt{45x^5}}{\sqrt{y^4}}\hfill \\ \begin{array}{c}\text{Simplify the radicals in the numerator and}\hfill \\ \text{the denominator.}\hfill \end{array}\hfill & & & \hfill \frac{\sqrt{9x^4}·\sqrt{5x}}{y^2}\hfill \\ \text{Simplify.}\hfill & & & \hfill \frac{3x^2\sqrt{5x}}{y^2}\hfill \end{array}$

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$\begin{array}{cccc}& & & \hfill \sqrt[3]{\frac{24x^7}{y^3}}\hfill \\ \\ \\ \begin{array}{c}\text{The fraction in the radicand cannot be}\hfill \\ \text{simplified. Use the Quotient Property to}\hfill \\ \text{write as two radicals.}\hfill \end{array}\hfill & & & \hfill \frac{\sqrt[3]{24x^7}}{\sqrt[3]{y^3}}\hfill \\ \\ \\ \begin{array}{c}\text{Rewrite each radicand as a product}\hfill \\ \text{using perfect cube factors.}\hfill \end{array}\hfill & & & \hfill \frac{\sqrt[3]{8x^6·3x}}{\sqrt[3]{y^3}}\hfill \\ \\ \\ \begin{array}{c}\text{Rewrite the numerator as the product of}\hfill \\ \text{two radicals.}\hfill \end{array}\hfill & & & \hfill \frac{\sqrt[3]{{\left(2x^2\right)}^3}·\sqrt[3]{3x}}{\sqrt[3]{y^3}}\hfill \\ \\ \\ \text{Simplify.}\hfill & & & \hfill \frac{2x^2\sqrt[3]{3x}}{y}\hfill \end{array}$

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$\begin{array}{cccc}& & & \hfill \sqrt[4]{\frac{48x^{10}}{y^8}}\hfill \\ \\ \\ \begin{array}{c}\text{The fraction in the radicand cannot be}\hfill \\ \text{simplified.}\hfill \end{array}\hfill & & & \hfill \frac{\sqrt[4]{48x^{10}}}{\sqrt[4]{y^8}}\hfill \\ \\ \\ \begin{array}{c}\text{Use the Quotient Property to write as two}\hfill \\ \text{radicals. Rewrite each radicand as a}\hfill \\ \text{product using perfect fourth power factors.}\hfill \end{array}\hfill & & & \hfill \frac{\sqrt[4]{16x^8·3x^2}}{\sqrt[4]{y^8}}\hfill \\ \\ \\ \begin{array}{c}\text{Rewrite the numerator as the product of}\hfill \\ \text{two radicals.}\hfill \end{array}\hfill & & & \hfill \frac{\sqrt[4]{{\left(2x^2\right)}^4}·\sqrt[4]{3x^2}}{\sqrt[4]{{\left(y^2\right)}^4}}\hfill \\ \\ \\ \text{Simplify.}\hfill & & & \hfill \frac{2x^2\sqrt[4]{3x^2}}{y^2}\hfill \end{array}$