9.5 Addition and subtraction of square root expressions

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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. The distinction between the principal square root of the number x and the secondary square root of the number x is made by explanation and by example. The simplification of the radical expressions that both involve and do not involve fractions is shown in many detailed examples; this is followed by an explanation of how and why radicals are eliminated from the denominator of a radical expression. Real-life applications of radical equations have been included, such as problems involving daily output, daily sales, electronic resonance frequency, and kinetic energy.Objectives of this module: understand the process used in adding and subtracting square roots, be able to add and subtract square roots.

Overview

• The Logic Behind The Process
• The Process

The logic behind the process

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

$\begin{array}{ccccc}4x+8x=12x\hfill & \hfill & \text{or}\hfill & \hfill & 4\sqrt{3}+8\sqrt{3}=12\sqrt{3}\hfill \\ \text{and}\hfill & \hfill & \hfill & \hfill & \hfill \\ 5a-11a+4\left(a+1\right)\hfill & \hfill & \text{or}\hfill & \hfill & 5\sqrt{2x}-11\sqrt{2x}+4\left(\sqrt{2x}+1\right)\hfill \\ 5a-11a+4a+4\hfill & \hfill & \hfill & \hfill & 5\sqrt{2x}-11\sqrt{2x}+4\sqrt{2x}+4\hfill \\ -2a+4\hfill & \hfill & \hfill & \hfill & -2\sqrt{2x}+4\hfill \end{array}$

The process

Let’s consider the expression $4\sqrt{3}+8\sqrt{3}.$ There are two ways to look at the simplification process:

1. We are asking, “How many square roots of 3 do we have?”

$4\sqrt{3}$ means we have 4 “square roots of 3”

$8\sqrt{3}$ means we have 8 “square roots of 3”

Thus, altogether we have 12 “square roots of 3.”
2. We can also use the idea of combining like terms. If we recall, the process of combining like terms is based on the distributive property

$\begin{array}{ccccc}4x+8x=12x& & \text{because}& & 4x+8x=\left(4+8\right)x=12x\end{array}$

We could simplify $4\sqrt{3}+8\sqrt{3}$ using the distributive property.

$4\sqrt{3}+8\sqrt{3}=\left(4+8\right)\sqrt{3}=12\sqrt{3}$

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.

Sample set a

$-6\sqrt{10}+11\sqrt{10}=5\sqrt{10}$

$\begin{array}{ccccc}4\sqrt{32}+5\sqrt{2}.\hfill & \hfill & \hfill & \hfill & \text{Simplify}\text{}\sqrt{32}.\hfill \\ 4\sqrt{16\text{}·\text{}2}+5\sqrt{2}\hfill & =\hfill & 4\sqrt{16}\sqrt{2}+5\sqrt{2}\hfill & \hfill & \hfill \\ \hfill & =\hfill & 4\text{}·\text{}4\sqrt{2}+5\sqrt{2}\hfill & \hfill & \hfill \\ \hfill & =\hfill & 16\sqrt{2}+5\sqrt{2}\hfill & \hfill & \hfill \\ \hfill & =\hfill & 21\sqrt{2}\hfill & \hfill & \hfill \end{array}$

$\begin{array}{ccccc}-3x\sqrt{75}+2x\sqrt{48}-x\sqrt{27}.\hfill & \hfill & \hfill & \hfill & \text{Simple each of the three radicals}\text{.}\hfill \\ \hfill & =\hfill & -3x\sqrt{25\text{}·\text{}3}+2x\sqrt{16\text{}·\text{}3}-x\sqrt{9\text{}·\text{}3}\hfill & \hfill & \hfill \\ \hfill & =\hfill & -15x\sqrt{3}+8x\sqrt{3}-3x\sqrt{3}\hfill & \hfill & \hfill \\ \hfill & =\hfill & \left(-15x+8x-3x\right)\sqrt{3}\hfill & \hfill & \hfill \\ \hfill & =\hfill & -10x\sqrt{3}\hfill & \hfill & \hfill \end{array}$

$\begin{array}{ccccc}5a\sqrt{24{a}^{3}}-7\sqrt{54{a}^{5}}+{a}^{2}\sqrt{6a}+6a.\hfill & \hfill & \hfill & \hfill & \text{Simplify each radical}\text{.}\hfill \\ \hfill & =\hfill & 5a\sqrt{4\text{}·\text{}6\text{}·\text{}{a}^{2}\text{}·\text{}a}-7\sqrt{9\text{}·\text{}6\text{}·\text{}{a}^{4}\text{}·\text{}a}+{a}^{2}\sqrt{6a}+6a\hfill & \hfill & \hfill \\ \hfill & =\hfill & 10{a}^{2}\sqrt{6a}-21{a}^{2}\sqrt{6a}+{a}^{2}\sqrt{6a}+6a\hfill & \hfill & \hfill \\ \hfill & =\hfill & \left(10{a}^{2}-21{a}^{2}+{a}^{2}\right)\sqrt{6a}+6a\hfill & \hfill & \hfill \\ \hfill & =\hfill & -10{a}^{2}\sqrt{6a}+6a\hfill & \hfill & \begin{array}{l}\text{Factor out}-2a\text{.}\hfill \\ \text{(This step is optional}\text{.)}\hfill \end{array}\hfill \\ \hfill & =\hfill & -2a\left(5a\sqrt{6a}-3\right)\hfill & \hfill & \hfill \end{array}$

Practice set a

Find each sum or difference.

$4\sqrt{18}-5\sqrt{8}$

$2\sqrt{2}$

$6x\sqrt{48}+8x\sqrt{75}$

$64x\sqrt{3}$

$-7\sqrt{84x}-12\sqrt{189x}+2\sqrt{21x}$

$-48\sqrt{21x}$

$9\sqrt{6}-8\sqrt{6}+3$

$\sqrt{6}+3$

$\sqrt{{a}^{3}}+4a\sqrt{a}$

$5a\sqrt{a}$

$4x\sqrt{54{x}^{3}}+\sqrt{36{x}^{2}}+3\sqrt{24{x}^{5}}-3x$

$18{x}^{2}\sqrt{6x}+3x$

Sample set b

$\begin{array}{ccccc}\frac{3+\sqrt{8}}{3-\sqrt{8}}.& & & & \begin{array}{l}\text{We'll\hspace{0.17em}rationalize\hspace{0.17em}the\hspace{0.17em}denominator\hspace{0.17em}by\hspace{0.17em}multiplying\hspace{0.17em}this\hspace{0.17em}fraction}\\ \text{by\hspace{0.17em}1\hspace{0.17em}in\hspace{0.17em}the\hspace{0.17em}form}\frac{3+\sqrt{8}}{3+\sqrt{8}}.\end{array}\\ \frac{3+\sqrt{8}}{3-\sqrt{8}}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{3+\sqrt{8}}{3+\sqrt{8}}& =& \frac{\left(3+\sqrt{8}\right)\left(3+\sqrt{8}\right)}{{3}^{2}-{\left(\sqrt{8}\right)}^{2}}& & \\ & =& \frac{9+3\sqrt{8}+3\sqrt{8}+\sqrt{8}\sqrt{8}}{9-8}& & \\ & =& \frac{9+6\sqrt{8}+8}{1}& & \\ & =& 17+6\sqrt{8}& & \\ & =& 17+6\sqrt{4\text{\hspace{0.17em}}·\text{\hspace{0.17em}}2}& & \\ & =& 17+12\sqrt{2}& & \end{array}$

$\begin{array}{ccccc}\frac{2+\sqrt{7}}{4-\sqrt{3}}.\hfill & \hfill & \hfill & \hfill & \begin{array}{l}\text{Rationalize the denominator by multiplying this fraction by}\hfill \\ \text{1 in the from}\frac{4+\sqrt{3}}{4+\sqrt{3}}.\hfill \end{array}\hfill \\ \frac{2+\sqrt{7}}{4-\sqrt{3}}\text{}·\text{}\frac{4+\sqrt{3}}{4+\sqrt{3}}\hfill & =\hfill & \frac{\left(2+\sqrt{7}\right)\left(4+\sqrt{3}\right)}{{4}^{2}-{\left(\sqrt{3}\right)}^{2}}\hfill & \hfill & \hfill \\ \hfill & =\hfill & \frac{8+2\sqrt{3}+4\sqrt{7}+\sqrt{21}}{16-3}\hfill & \hfill & \hfill \\ \hfill & =\hfill & \frac{8+2\sqrt{3}+4\sqrt{7}+\sqrt{21}}{13}\hfill & \hfill & \hfill \end{array}$

Practice set b

Simplify each by performing the indicated operation.

$\sqrt{5}\left(\sqrt{6}-4\right)$

$\sqrt{30}-4\sqrt{5}$

$\left(\sqrt{5}+\sqrt{7}\right)\left(\sqrt{2}+\sqrt{8}\right)$

$3\sqrt{10}+3\sqrt{14}$

$\left(3\sqrt{2}-2\sqrt{3}\right)\left(4\sqrt{3}+\sqrt{8}\right)$

$8\sqrt{6}-12$

$\frac{4+\sqrt{5}}{3-\sqrt{8}}$

$12+8\sqrt{2}+3\sqrt{5}+2\sqrt{10}$

Exercises

For the following problems, simplify each expression by performing the indicated operation.

$4\sqrt{5}-2\sqrt{5}$

$2\sqrt{5}$

$10\sqrt{2}+8\sqrt{2}$

$-3\sqrt{6}-12\sqrt{6}$

$-15\sqrt{6}$

$-\sqrt{10}-2\sqrt{10}$

$3\sqrt{7x}+2\sqrt{7x}$

$5\sqrt{7x}$

$6\sqrt{3a}+\sqrt{3a}$

$2\sqrt{18}+5\sqrt{32}$

$26\sqrt{2}$

$4\sqrt{27}-3\sqrt{48}$

$\sqrt{200}-\sqrt{128}$

$2\sqrt{2}$

$4\sqrt{300}+2\sqrt{500}$

$6\sqrt{40}+8\sqrt{80}$

$12\sqrt{10}+32\sqrt{5}$

$2\sqrt{120}-5\sqrt{30}$

$8\sqrt{60}-3\sqrt{15}$

$13\sqrt{15}$

$\sqrt{{a}^{3}}-3a\sqrt{a}$

$\sqrt{4{x}^{3}}+x\sqrt{x}$

$3x\sqrt{x}$

$2b\sqrt{{a}^{3}{b}^{5}}+6a\sqrt{a{b}^{7}}$

$5xy\sqrt{2x{y}^{3}}-3{y}^{2}\sqrt{2{x}^{3}y}$

$2x{y}^{2}\sqrt{2xy}$

$5\sqrt{20}+3\sqrt{45}-3\sqrt{40}$

$\sqrt{24}-2\sqrt{54}-4\sqrt{12}$

$-4\sqrt{6}-8\sqrt{3}$

$6\sqrt{18}+5\sqrt{32}+4\sqrt{50}$

$-8\sqrt{20}-9\sqrt{125}+10\sqrt{180}$

$-\sqrt{5}$

$2\sqrt{27}+4\sqrt{3}-6\sqrt{12}$

$\sqrt{14}+2\sqrt{56}-3\sqrt{136}$

$5\sqrt{14}-6\sqrt{34}$

$3\sqrt{2}+2\sqrt{63}+5\sqrt{7}$

$4ax\sqrt{3x}+2\sqrt{3{a}^{2}{x}^{3}}+7\sqrt{3{a}^{2}{x}^{3}}$

$13ax\sqrt{3x}$

$3by\sqrt{5y}+4\sqrt{5{b}^{2}{y}^{3}}-2\sqrt{5{b}^{2}{y}^{3}}$

$\sqrt{2}\left(\sqrt{3}+1\right)$

$\sqrt{6}+\sqrt{2}$

$\sqrt{3}\left(\sqrt{5}-3\right)$

$\sqrt{5}\left(\sqrt{3}-\sqrt{2}\right)$

$\sqrt{15}-\sqrt{10}$

$\sqrt{7}\left(\sqrt{6}-\sqrt{3}\right)$

$\sqrt{8}\left(\sqrt{3}+\sqrt{2}\right)$

$2\left(\sqrt{6}+2\right)$

$\sqrt{10}\left(\sqrt{10}-\sqrt{5}\right)$

$\left(1+\sqrt{3}\right)\left(2-\sqrt{3}\right)$

$-1+\sqrt{3}$

$\left(5+\sqrt{6}\right)\left(4-\sqrt{6}\right)$

$\left(3-\sqrt{2}\right)\left(4-\sqrt{2}\right)$

$7\left(2-\sqrt{2}\right)$

$\left(5+\sqrt{7}\right)\left(4-\sqrt{7}\right)$

$\left(\sqrt{2}+\sqrt{5}\right)\left(\sqrt{2}+3\sqrt{5}\right)$

$17+4\sqrt{10}$

$\left(2\sqrt{6}-\sqrt{3}\right)\left(3\sqrt{6}+2\sqrt{3}\right)$

$\left(4\sqrt{5}-2\sqrt{3}\right)\left(3\sqrt{5}+\sqrt{3}\right)$

$54-2\sqrt{15}$

$\left(3\sqrt{8}-2\sqrt{2}\right)\left(4\sqrt{2}-5\sqrt{8}\right)$

$\left(\sqrt{12}+5\sqrt{3}\right)\left(2\sqrt{3}-2\sqrt{12}\right)$

$-42$

${\left(1+\sqrt{3}\right)}^{2}$

${\left(3+\sqrt{5}\right)}^{2}$

$14+6\sqrt{5}$

${\left(2-\sqrt{6}\right)}^{2}$

${\left(2-\sqrt{7}\right)}^{2}$

$11-4\sqrt{7}$

${\left(1+\sqrt{3x}\right)}^{2}$

${\left(2+\sqrt{5x}\right)}^{2}$

$4+4\sqrt{5x}+5x$

${\left(3-\sqrt{3x}\right)}^{2}$

${\left(8-\sqrt{6b}\right)}^{2}$

$64-16\sqrt{6b}+6b$

${\left(2a+\sqrt{5a}\right)}^{2}$

${\left(3y-\sqrt{7y}\right)}^{2}$

$9{y}^{2}-6y\sqrt{7y}+7y$

$\left(3+\sqrt{3}\right)\left(3-\sqrt{3}\right)$

$\left(2+\sqrt{5}\right)\left(2-\sqrt{5}\right)$

$-1$

$\left(8+\sqrt{10}\right)\left(8-\sqrt{10}\right)$

$\left(6+\sqrt{7}\right)\left(6-\sqrt{7}\right)$

29

$\left(\sqrt{2}+\sqrt{3}\right)\left(\sqrt{2}-\sqrt{3}\right)$

$\left(\sqrt{5}+\sqrt{2}\right)\left(\sqrt{5}-\sqrt{2}\right)$

3

$\left(\sqrt{a}+\sqrt{b}\right)\left(\sqrt{a}-\sqrt{b}\right)$

$\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}\right)$

$x-y$

$\frac{2}{5+\sqrt{3}}$

$\frac{4}{6+\sqrt{2}}$

$\frac{2\left(6-\sqrt{2}\right)}{17}$

$\frac{1}{3-\sqrt{2}}$

$\frac{1}{4-\sqrt{3}}$

$\frac{4+\sqrt{3}}{13}$

$\frac{8}{2-\sqrt{6}}$

$\frac{2}{3-\sqrt{7}}$

$3+\sqrt{7}$

$\frac{\sqrt{5}}{3+\sqrt{3}}$

$\frac{\sqrt{3}}{6+\sqrt{6}}$

$\frac{2\sqrt{3}-\sqrt{2}}{10}$

$\frac{2-\sqrt{8}}{2+\sqrt{8}}$

$\frac{4+\sqrt{5}}{4-\sqrt{5}}$

$\frac{21+8\sqrt{5}}{11}$

$\frac{1+\sqrt{6}}{1-\sqrt{6}}$

$\frac{8-\sqrt{3}}{2+\sqrt{18}}$

$\frac{-16+2\sqrt{3}+24\sqrt{2}-3\sqrt{6}}{14}$

$\frac{6-\sqrt{2}}{4+\sqrt{12}}$

$\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}}$

$5-2\sqrt{6}$

$\frac{\sqrt{6a}-\sqrt{8a}}{\sqrt{8a}+\sqrt{6a}}$

$\frac{\sqrt{2b}-\sqrt{3b}}{\sqrt{3b}+\sqrt{2b}}$

$2\sqrt{6}-5$

Exercises for review

( [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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