8.4 Add, Subtract, and Multiply Radical Expressions

ⓐ
$\begin{array}{cccc}& & & \hfill 2\sqrt{2}-7\sqrt{2}\hfill \\ \begin{array}{c}\text{Since the radicals are like, we subtract the}\hfill \\ \text{coefficients.}\hfill \end{array}\hfill & & & \hfill \mathrm{-5}\sqrt{2}\hfill \end{array}$
ⓑ
$\begin{array}{cccc}& & & \hfill 5\sqrt[3]{y}+4\sqrt[3]{y}\hfill \\ \begin{array}{c}\text{Since the radicals are like, we add the}\hfill \\ \text{coefficients.}\hfill \end{array}\hfill & & & \hfill 9\sqrt[3]{y}\hfill \end{array}$

ⓒ
$\begin{array}{cccc}& & & \hfill 7\sqrt[4]{x}-2\sqrt[4]{y}\hfill \end{array}$

The indices are the same but the radicals are different. These are not like radicals. Since the radicals are not like, we cannot subtract them.

ⓐ
$\begin{array}{cccc}& & & \hfill 2\sqrt{5n}-6\sqrt{5n}+4\sqrt{5n}\hfill \\ \text{Since the radicals are like, we combine them.}\hfill & & & \hfill 0\sqrt{5n}\hfill \\ \text{Simplify.}\hfill & & & \hfill 0\hfill \end{array}$

ⓑ
$\begin{array}{cccc}& & & \hfill \sqrt[4]{3xy}+5\sqrt[4]{3xy}-4\sqrt[4]{3xy}\hfill \\ \text{Since the radicals are like, we combine them.}\hfill & & & \hfill 2\sqrt[4]{3xy}\hfill \end{array}$

ⓐ
$\begin{array}{cccc}& & & \hfill \sqrt{20}+3\sqrt{5}\hfill \\ \text{Simplify the radicals, when possible.}\hfill & & & \hfill \sqrt{4}·\sqrt{5}+3\sqrt{5}\hfill \\ & & & \hfill 2\sqrt{5}+3\sqrt{5}\hfill \\ \text{Combine the like radicals.}\hfill & & & \hfill 5\sqrt{5}\hfill \end{array}$

ⓑ
$\begin{array}{cccc}& & & \hfill \sqrt[3]{24}-\sqrt[3]{375}\hfill \\ \text{Simplify the radicals.}\hfill & & & \hfill \sqrt[3]{8}·\sqrt[3]{3}-\sqrt[3]{125}·\sqrt[3]{3}\hfill \\ & & & \hfill 2\sqrt[3]{3}-5\sqrt[3]{3}\hfill \\ \text{Combine the like radicals.}\hfill & & & \hfill \mathrm{-3}\sqrt[3]{3}\hfill \end{array}$

ⓒ
$\begin{array}{cccc}& & & \hfill \frac{1}{2}\sqrt[4]{48}-\frac{2}{3}\sqrt[4]{243}\hfill \\ \text{Simplify the radicals.}\hfill & & & \hfill \frac{1}{2}\sqrt[4]{16}·\sqrt[4]{3}-\frac{2}{3}\sqrt[4]{81}·\sqrt[4]{3}\hfill \\ & & & \hfill \frac{1}{2}·2·\sqrt[4]{3}-\frac{2}{3}·3·\sqrt[4]{3}\hfill \\ & & & \hfill \sqrt[4]{3}-2\sqrt[4]{3}\hfill \\ \text{Combine the like radicals.}\hfill & & & \hfill \text{−}\sqrt[4]{3}\hfill \end{array}$

ⓐ
$\begin{array}{cccc}& & & \hfill 9\sqrt{50m^2}-6\sqrt{48m^2}\hfill \\ \text{Simplify the radicals.}\hfill & & & \hfill 9\sqrt{25m^2}·\sqrt{2}-6\sqrt{16m^2}·\sqrt{3}\hfill \\ & & & \hfill 9·5m·\sqrt{2}-6·4m·\sqrt{3}\hfill \\ & & & \hfill 45m\sqrt{2}-24m\sqrt{3}\hfill \\ \begin{array}{c}\text{The radicals are not like and so cannot be}\hfill \\ \text{combined.}\hfill \end{array}\hfill & & & \end{array}$

ⓑ
$\begin{array}{cccc}& & & \hfill \sqrt[3]{54n^5}-\sqrt[3]{16n^5}\hfill \\ \text{Simplify the radicals.}\hfill & & & \hfill \sqrt[3]{27n^3}·\sqrt[3]{2n^2}-\sqrt[3]{8n^3}·\sqrt[3]{2n^2}\hfill \\ & & & \hfill 3n\sqrt[3]{2n^2}-2n\sqrt[3]{2n^2}\hfill \\ \text{Combine the like radicals.}\hfill & & & \hfill n\sqrt[3]{2n^2}\hfill \end{array}$

ⓐ
$\begin{array}{cccc}& & & \hfill \left(6\sqrt{2}\right)\left(3\sqrt{10}\right)\hfill \\ \text{Multiply using the Product Property.}\hfill & & & \hfill 18\sqrt{20}\hfill \\ \text{Simplify the radical.}\hfill & & & \hfill 18\sqrt{4}·\sqrt{5}\hfill \\ \text{Simplify.}\hfill & & & \hfill 18·2·\sqrt{5}\hfill \\ & & & \hfill 36\sqrt{5}\hfill \end{array}$

ⓑ
$\begin{array}{cccc}& & & \hfill \left(\mathrm{-5}\sqrt[3]{4}\right)\left(\mathrm{-4}\sqrt[3]{6}\right)\hfill \\ \text{Multiply using the Product Property.}\hfill & & & \hfill 20\sqrt[3]{24}\hfill \\ \text{Simplify the radical.}\hfill & & & \hfill 20\sqrt[3]{8}·\sqrt[3]{3}\hfill \\ \text{Simplify.}\hfill & & & \hfill 20·2·\sqrt[3]{3}\hfill \\ & & & \hfill 40\sqrt[3]{3}\hfill \end{array}$

ⓐ
$\begin{array}{cccc}& & & \hfill \left(10\sqrt{6p^3}\right)\left(4\sqrt{3p}\right)\hfill \\ \text{Multiply.}\hfill & & & \hfill 40\sqrt{18p^4}\hfill \\ \text{Simplify the radical.}\hfill & & & \hfill 40\sqrt{9p^4}·\sqrt{2}\hfill \\ \text{Simplify.}\hfill & & & \hfill 40·3p^2·\sqrt{3}\hfill \\ & & & \hfill 120p^2\sqrt{3}\hfill \end{array}$

ⓑ When the radicands involve large numbers, it is often advantageous to factor them in order to find the perfect powers.

$\begin{array}{cccc}& & & \hfill \left(2\sqrt[4]{20y^2}\right)\left(3\sqrt[4]{28y^3}\right)\hfill \\ \text{Multiply.}\hfill & & & \hfill 6\sqrt[4]{4·5·4·7y^5}\hfill \\ \text{Simplify the radical.}\hfill & & & \hfill 6\sqrt[4]{16y^4}·\sqrt[4]{35y^{}}\hfill \\ \text{Simplify.}\hfill & & & \hfill 6·2y\sqrt[4]{35y^{}}\hfill \\ \text{Multiply.}\hfill & & & \hfill 12y\sqrt[4]{35y^{}}\hfill \end{array}$

ⓐ
$\begin{array}{cccc}& & & \hfill \sqrt{6}\left(\sqrt{2}+\sqrt{18}\right)\hfill \\ \text{Multiply.}\hfill & & & \hfill \sqrt{12}+\sqrt{108}\hfill \\ \text{Simplify.}\hfill & & & \hfill \sqrt{4}·\sqrt{3}+\sqrt{36}·\sqrt{3}\hfill \\ \text{Simplify.}\hfill & & & \hfill 2\sqrt{3}+6\sqrt{3}\hfill \\ \text{Combine like radicals.}\hfill & & & \hfill 8\sqrt{3}\hfill \end{array}$

ⓑ
$\begin{array}{cccc}& & & \hfill \sqrt[3]{9}\left(5-\sqrt[3]{18}\right)\hfill \\ \text{Distribute.}\hfill & & & \hfill 5\sqrt[3]{9}-\sqrt[3]{162}\hfill \\ \text{Simplify.}\hfill & & & \hfill 5\sqrt[3]{9}-\sqrt[3]{27}·\sqrt[3]{6}\hfill \\ \text{Simplify.}\hfill & & & \hfill 5\sqrt[3]{9}-3\sqrt[3]{6}\hfill \end{array}$

ⓐ
$\begin{array}{cccc}& & & \hfill \left(3-2\sqrt{7}\right)\left(4-2\sqrt{7}\right)\hfill \\ \text{Multiply}\hfill & & & \hfill 12-6\sqrt{7}-8\sqrt{7}+4·7\hfill \\ \text{Simplify.}\hfill & & & \hfill 12-6\sqrt{7}-8\sqrt{7}+28\hfill \\ \text{Combine like terms.}\hfill & & & \hfill 40-14\sqrt{7}\hfill \end{array}$

ⓑ
$\begin{array}{cccc}& & & \hfill \left(\sqrt[3]{x}-2\right)\left(\sqrt[3]{x}+4\right)\hfill \\ \text{Multiply.}\hfill & & & \hfill \sqrt[3]{x^2}+4\sqrt[3]{x}-2\sqrt[3]{x}-8\hfill \\ \text{Combine like terms.}\hfill & & & \hfill \sqrt[3]{x^2}+2\sqrt[3]{x}-8\hfill \end{array}$

$\begin{array}{cccc}& & & \hfill \left(3\sqrt{2}-\sqrt{5}\right)\left(\sqrt{2}+4\sqrt{5}\right)\hfill \\ \text{Multiply.}\hfill & & & \hfill 3·2+12\sqrt{10}-\sqrt{10}-4·5\hfill \\ \text{Simplify.}\hfill & & & \hfill 6+12\sqrt{10}-\sqrt{10}-20\hfill \\ \text{Combine like terms.}\hfill & & & \hfill \mathrm{-14}+11\sqrt{10}\hfill \end{array}$

Be sure to include the $2ab$ term when squaring a binomial.

ⓐ

Multiply, using the Product of Binomial Squares Pattern.
Simplify.
Combine like terms.

ⓑ

Multiply, using the Product of Binomial Squares Pattern.
Simplify.
Combine like terms.