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Now we will study methods of simplifying radical expressions such as
$\begin{array}{ccccc}4\sqrt{3}+8\sqrt{3}& & \text{or}& & 5\sqrt{2x}-11\sqrt{2x}+4\left(\sqrt{2x}+1\right)\end{array}$
The procedure for adding and subtracting square root expressions will become apparent if we think back to the procedure we used for simplifying polynomial expressions such as
$\begin{array}{ccccc}4x+8x& & \text{or}& & 5a-11a+4\left(a+1\right)\end{array}$
The variables
$x$ and
$a$ are letters representing some unknown quantities (perhaps
$x$ represents
$\sqrt{3}$ and
$a$ represents
$\sqrt{2x}$ ). Combining like terms gives us
$\begin{array}{ccccc}4x+8x=12x\hfill & \hfill & \text{or}\hfill & \hfill & 4\sqrt{3}+8\sqrt{3}=12\sqrt{3}\hfill \\ \text{and}\hfill & \hfill & \hfill & \hfill & \hfill \\ 5a-11a+4\left(a+1\right)\hfill & \hfill & \text{or}\hfill & \hfill & 5\sqrt{2x}-11\sqrt{2x}+4\left(\sqrt{2x}+1\right)\hfill \\ 5a-11a+4a+4\hfill & \hfill & \hfill & \hfill & 5\sqrt{2x}-11\sqrt{2x}+4\sqrt{2x}+4\hfill \\ -2a+4\hfill & \hfill & \hfill & \hfill & -2\sqrt{2x}+4\hfill \end{array}$
Let’s consider the expression $4\sqrt{3}+8\sqrt{3}.$ There are two ways to look at the simplification process:
Both methods will give us the same result. The first method is probably a bit quicker, but keep in mind, however, that the process works because it is based on one of the basic rules of algebra, the distributive property of real numbers.
Simplify the following radical expressions.
$-6\sqrt{10}+11\sqrt{10}=5\sqrt{10}$
$\begin{array}{ccccc}4\sqrt{32}+5\sqrt{2}.\hfill & \hfill & \hfill & \hfill & \text{Simplify}\text{}\sqrt{32}.\hfill \\ 4\sqrt{16\text{}\xb7\text{}2}+5\sqrt{2}\hfill & =\hfill & 4\sqrt{16}\sqrt{2}+5\sqrt{2}\hfill & \hfill & \hfill \\ \hfill & =\hfill & 4\text{}\xb7\text{}4\sqrt{2}+5\sqrt{2}\hfill & \hfill & \hfill \\ \hfill & =\hfill & 16\sqrt{2}+5\sqrt{2}\hfill & \hfill & \hfill \\ \hfill & =\hfill & 21\sqrt{2}\hfill & \hfill & \hfill \end{array}$
$\begin{array}{ccccc}-3x\sqrt{75}+2x\sqrt{48}-x\sqrt{27}.\hfill & \hfill & \hfill & \hfill & \text{Simple each of the three radicals}\text{.}\hfill \\ \hfill & =\hfill & -3x\sqrt{25\text{}\xb7\text{}3}+2x\sqrt{16\text{}\xb7\text{}3}-x\sqrt{9\text{}\xb7\text{}3}\hfill & \hfill & \hfill \\ \hfill & =\hfill & -15x\sqrt{3}+8x\sqrt{3}-3x\sqrt{3}\hfill & \hfill & \hfill \\ \hfill & =\hfill & \left(-15x+8x-3x\right)\sqrt{3}\hfill & \hfill & \hfill \\ \hfill & =\hfill & -10x\sqrt{3}\hfill & \hfill & \hfill \end{array}$
$\begin{array}{ccccc}5a\sqrt{24{a}^{3}}-7\sqrt{54{a}^{5}}+{a}^{2}\sqrt{6a}+6a.\hfill & \hfill & \hfill & \hfill & \text{Simplify each radical}\text{.}\hfill \\ \hfill & =\hfill & 5a\sqrt{4\text{}\xb7\text{}6\text{}\xb7\text{}{a}^{2}\text{}\xb7\text{}a}-7\sqrt{9\text{}\xb7\text{}6\text{}\xb7\text{}{a}^{4}\text{}\xb7\text{}a}+{a}^{2}\sqrt{6a}+6a\hfill & \hfill & \hfill \\ \hfill & =\hfill & 10{a}^{2}\sqrt{6a}-21{a}^{2}\sqrt{6a}+{a}^{2}\sqrt{6a}+6a\hfill & \hfill & \hfill \\ \hfill & =\hfill & \left(10{a}^{2}-21{a}^{2}+{a}^{2}\right)\sqrt{6a}+6a\hfill & \hfill & \hfill \\ \hfill & =\hfill & -10{a}^{2}\sqrt{6a}+6a\hfill & \hfill & \begin{array}{l}\text{Factor out}-2a\text{.}\hfill \\ \text{(This step is optional}\text{.)}\hfill \end{array}\hfill \\ \hfill & =\hfill & -2a\left(5a\sqrt{6a}-3\right)\hfill & \hfill & \hfill \end{array}$
Find each sum or difference.
$4x\sqrt{54{x}^{3}}+\sqrt{36{x}^{2}}+3\sqrt{24{x}^{5}}-3x$
$18{x}^{2}\sqrt{6x}+3x$
$\begin{array}{ccccc}\frac{3+\sqrt{8}}{3-\sqrt{8}}.& & & & \begin{array}{l}\text{We'll\hspace{0.17em}rationalize\hspace{0.17em}the\hspace{0.17em}denominator\hspace{0.17em}by\hspace{0.17em}multiplying\hspace{0.17em}this\hspace{0.17em}fraction}\\ \text{by\hspace{0.17em}1\hspace{0.17em}in\hspace{0.17em}the\hspace{0.17em}form}\frac{3+\sqrt{8}}{3+\sqrt{8}}.\end{array}\\ \frac{3+\sqrt{8}}{3-\sqrt{8}}\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}\frac{3+\sqrt{8}}{3+\sqrt{8}}& =& \frac{(3+\sqrt{8})(3+\sqrt{8})}{{3}^{2}-{\left(\sqrt{8}\right)}^{2}}& & \\ & =& \frac{9+3\sqrt{8}+3\sqrt{8}+\sqrt{8}\sqrt{8}}{9-8}& & \\ & =& \frac{9+6\sqrt{8}+8}{1}& & \\ & =& 17+6\sqrt{8}& & \\ & =& 17+6\sqrt{4\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}2}& & \\ & =& 17+12\sqrt{2}& & \end{array}$
$\begin{array}{ccccc}\frac{2+\sqrt{7}}{4-\sqrt{3}}.\hfill & \hfill & \hfill & \hfill & \begin{array}{l}\text{Rationalize the denominator by multiplying this fraction by}\hfill \\ \text{1 in the from}\frac{4+\sqrt{3}}{4+\sqrt{3}}.\hfill \end{array}\hfill \\ \frac{2+\sqrt{7}}{4-\sqrt{3}}\text{}\xb7\text{}\frac{4+\sqrt{3}}{4+\sqrt{3}}\hfill & =\hfill & \frac{\left(2+\sqrt{7}\right)\left(4+\sqrt{3}\right)}{{4}^{2}-{\left(\sqrt{3}\right)}^{2}}\hfill & \hfill & \hfill \\ \hfill & =\hfill & \frac{8+2\sqrt{3}+4\sqrt{7}+\sqrt{21}}{16-3}\hfill & \hfill & \hfill \\ \hfill & =\hfill & \frac{8+2\sqrt{3}+4\sqrt{7}+\sqrt{21}}{13}\hfill & \hfill & \hfill \end{array}$
Simplify each by performing the indicated operation.
$\left(\sqrt{5}+\sqrt{7}\right)\left(\sqrt{2}+\sqrt{8}\right)$
$3\sqrt{10}+3\sqrt{14}$
$\left(3\sqrt{2}-2\sqrt{3}\right)\left(4\sqrt{3}+\sqrt{8}\right)$
$8\sqrt{6}-12$
$\frac{4+\sqrt{5}}{3-\sqrt{8}}$
$12+8\sqrt{2}+3\sqrt{5}+2\sqrt{10}$
For the following problems, simplify each expression by performing the indicated operation.
$10\sqrt{2}+8\sqrt{2}$
$-\sqrt{10}-2\sqrt{10}$
$6\sqrt{3a}+\sqrt{3a}$
$4\sqrt{27}-3\sqrt{48}$
$4\sqrt{300}+2\sqrt{500}$
$2\sqrt{120}-5\sqrt{30}$
$\sqrt{{a}^{3}}-3a\sqrt{a}$
$2b\sqrt{{a}^{3}{b}^{5}}+6a\sqrt{a{b}^{7}}$
$5xy\sqrt{2x{y}^{3}}-3{y}^{2}\sqrt{2{x}^{3}y}$
$2x{y}^{2}\sqrt{2xy}$
$5\sqrt{20}+3\sqrt{45}-3\sqrt{40}$
$6\sqrt{18}+5\sqrt{32}+4\sqrt{50}$
$2\sqrt{27}+4\sqrt{3}-6\sqrt{12}$
$3\sqrt{2}+2\sqrt{63}+5\sqrt{7}$
$4ax\sqrt{3x}+2\sqrt{3{a}^{2}{x}^{3}}+7\sqrt{3{a}^{2}{x}^{3}}$
$13ax\sqrt{3x}$
$3by\sqrt{5y}+4\sqrt{5{b}^{2}{y}^{3}}-2\sqrt{5{b}^{2}{y}^{3}}$
$\sqrt{3}\left(\sqrt{5}-3\right)$
$\sqrt{5}\left(\sqrt{3}-\sqrt{2}\right)$
$\sqrt{15}-\sqrt{10}$
$\sqrt{7}\left(\sqrt{6}-\sqrt{3}\right)$
$\sqrt{8}\left(\sqrt{3}+\sqrt{2}\right)$
$2\left(\sqrt{6}+2\right)$
$\sqrt{10}\left(\sqrt{10}-\sqrt{5}\right)$
$\left(1+\sqrt{3}\right)\left(2-\sqrt{3}\right)$
$-1+\sqrt{3}$
$\left(5+\sqrt{6}\right)\left(4-\sqrt{6}\right)$
$\left(3-\sqrt{2}\right)\left(4-\sqrt{2}\right)$
$7\left(2-\sqrt{2}\right)$
$\left(5+\sqrt{7}\right)\left(4-\sqrt{7}\right)$
$\left(\sqrt{2}+\sqrt{5}\right)\left(\sqrt{2}+3\sqrt{5}\right)$
$17+4\sqrt{10}$
$\left(2\sqrt{6}-\sqrt{3}\right)\left(3\sqrt{6}+2\sqrt{3}\right)$
$\left(4\sqrt{5}-2\sqrt{3}\right)\left(3\sqrt{5}+\sqrt{3}\right)$
$54-2\sqrt{15}$
$\left(3\sqrt{8}-2\sqrt{2}\right)\left(4\sqrt{2}-5\sqrt{8}\right)$
$\left(\sqrt{12}+5\sqrt{3}\right)\left(2\sqrt{3}-2\sqrt{12}\right)$
$-42$
${\left(1+\sqrt{3}\right)}^{2}$
${\left(2-\sqrt{6}\right)}^{2}$
${\left(1+\sqrt{3x}\right)}^{2}$
${\left(3-\sqrt{3x}\right)}^{2}$
${\left(2a+\sqrt{5a}\right)}^{2}$
$\left(3+\sqrt{3}\right)\left(3-\sqrt{3}\right)$
$\left(8+\sqrt{10}\right)\left(8-\sqrt{10}\right)$
$\left(\sqrt{2}+\sqrt{3}\right)\left(\sqrt{2}-\sqrt{3}\right)$
$\left(\sqrt{5}+\sqrt{2}\right)\left(\sqrt{5}-\sqrt{2}\right)$
3
$\left(\sqrt{a}+\sqrt{b}\right)\left(\sqrt{a}-\sqrt{b}\right)$
$\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}\right)$
$x-y$
$\frac{2}{5+\sqrt{3}}$
$\frac{4}{6+\sqrt{2}}$
$\frac{2\left(6-\sqrt{2}\right)}{17}$
$\frac{1}{3-\sqrt{2}}$
$\frac{8}{2-\sqrt{6}}$
$\frac{\sqrt{5}}{3+\sqrt{3}}$
$\frac{\sqrt{3}}{6+\sqrt{6}}$
$\frac{2\sqrt{3}-\sqrt{2}}{10}$
$\frac{2-\sqrt{8}}{2+\sqrt{8}}$
$\frac{1+\sqrt{6}}{1-\sqrt{6}}$
$\frac{8-\sqrt{3}}{2+\sqrt{18}}$
$\frac{-16+2\sqrt{3}+24\sqrt{2}-3\sqrt{6}}{14}$
$\frac{6-\sqrt{2}}{4+\sqrt{12}}$
$\frac{\sqrt{6a}-\sqrt{8a}}{\sqrt{8a}+\sqrt{6a}}$
$\frac{\sqrt{2b}-\sqrt{3b}}{\sqrt{3b}+\sqrt{2b}}$
$2\sqrt{6}-5$
( [link] ) Simplify ${\left(\frac{{x}^{5}{y}^{3}}{{x}^{2}y}\right)}^{5}.$
( [link] ) Simplify ${\left(8{x}^{3}y\right)}^{2}{\left({x}^{2}{y}^{3}\right)}^{4}.$
$64{x}^{14}{y}^{14}$
( [link] ) Write ${\left(x-1\right)}^{4}{\left(x-1\right)}^{-7}$ so that only positive exponents appear.
( [link] ) Simplify $\sqrt{27{x}^{5}{y}^{10}{z}^{3}.}$
$3{x}^{2}{y}^{5}z\sqrt{3xz}$
( [link] ) Simplify $\frac{1}{2+\sqrt{x}}$ by rationalizing the denominator.
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