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9.6: Addition and Subtraction of Square Root Expressions

  • Page ID
    49398
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    The Logic Behind The Process

    Now we will study methods of simplifying radical expressions such as

    \(4\sqrt{3} + 8\sqrt{3}\) or \(5\sqrt{2x} - 11\sqrt{2x} + 4(\sqrt{2x} + 1)\)

    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
    \(4x + 8x\) or \(5a - 11a + 4(a+1)\)

    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

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

    The Process

    Let's consider the expression \(4\sqrt{3} + 8\sqrt{3}\). There are two ways to look at the simplification process.

    Simplification Process

    We are asking, "How many square roots of \(3\) do we have?"

    \(4 \sqrt{3}\) means we have \(4\) "square roots of \(3\)"

    Thus, altogether we have \(12\) "square roots of \(3\)."

    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

    \(4x + 8x = 12x\) because \(4x + 8x = (4 + 8)x = 12x\)

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

    4\sqrt{3} + 8\sqrt{3} = (4 + 8)\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

    Simplify the following radical expressions.

    Example \(\PageIndex{1}\)

    \(-6\sqrt{10} + 11\sqrt{10} = 5\sqrt{10}\)

    Example \(\PageIndex{2}\)

    \(4\sqrt{32} + 5\sqrt{2} \text{ Simplify } \sqrt{32}\)

    \(\begin{array}{flushleft}
    4\sqrt{16 \cdot 2} + 5\sqrt{2} &= 4\sqrt{16}\sqrt{2} + 5\sqrt{2}\\
    &= 4 \cdot 4\sqrt{2} + 5\sqrt{2}\\
    &= 16\sqrt{2} + 5\sqrt{2}\\
    &=21\sqrt{2}
    \end{array}\)

    Example \(\PageIndex{3}\)

    \(-3x\sqrt{75} + 2x\sqrt{48} - x\sqrt{27} \text{ Simplify each of the three radicals}\)

    \(\begin{array}{flushleft}
    -3x\sqrt{75} + 2x\sqrt{48} - x\sqrt{27} &= -3x\sqrt{25 \cdot 3} + 2x\sqrt{16 \cdot 3} - x\sqrt{9 \cdot 2}\\
    &= -15x\sqrt{3} + 8x\sqrt{3} - 3x\sqrt{3}\\
    &=(-15x + 8x - 3x)\sqrt{3}\\
    &=-10x\sqrt{3}
    \end{array}\)

    Example \(\PageIndex{4}\)

    \(5a\sqrt{24a^3} - 7\sqrt{54a^5} + a^2\sqrt{6a} + 6a \text{ Simplify each radical}\)

    \(\begin{array}{flushleft}
    5a\sqrt{24a^3} - 7\sqrt{54a^5} + a^2\sqrt{6a} + 6a &= 5a\sqrt{4 \cdot 6 \cdot a^2 \cdot a} - 7\sqrt{9 \cdot 6 \cdot a^4 \cdot a} + a^2\sqrt{6a} + 6a\\
    &= 10a^2\sqrt{6a} - 21a^2\sqrt{6a} + a^2\sqrt{6a} + 6a\\
    &= (10a^2 - 21a^2 + a^2) \sqrt{6a} + 6a\\
    &= -10a^2\sqrt{6a} + 6a\\
    &= -2a(5a\sqrt{6a} - 3)
    \end{array}\)

    Practice Set A

    Find each sum or difference.

    Practice Problem \(\PageIndex{1}\)

    \(4\sqrt{18} - 5\sqrt{8}\)

    Answer

    \(2\sqrt{2}\)

    Practice Problem \(\PageIndex{2}\)

    \(6x\sqrt{48} + 8x\sqrt{75}\)

    Answer

    \(64x\sqrt{3}\)

    Practice Problem \(\PageIndex{3}\)

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

    Answer

    \(-48\sqrt{21x}\)

    Practice Problem \(\PageIndex{4}\)

    \(9\sqrt{6} - 8\sqrt{6} + 3\)

    Answer

    \(\sqrt{6} + 3\)

    Practice Problem \(\PageIndex{5}\)

    \(\sqrt{a^3} + 4a\sqrt{a}\)

    Answer

    \(5a\sqrt{a}\)

    Practice Problem \(\PageIndex{6}\)

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

    Answer

    \(18x^2\sqrt{6x} + 3x\)

    Sample Set B

    Example \(\PageIndex{5}\)

    Finding the product of the square root of seven and the binomial the square root of eight minus three, using the rule for multiplying square root expressions. See the longdesc for a full description.

    Example \(\PageIndex{6}\)

    Finding the product of the binomial the square root of two plus the square root of three and the binomial the square root of five plus the square root of twelve, using the rule for multiplying square root expressions. See the longdesc for a full description.

    Example \(\PageIndex{7}\)

    Finding the product of the binomial four times the square root of two minus three times the square root of six and the binomial five times the square root of two plus the square root of six, using the rule for multiplying square root expressions. See the longdesc for a full description.

    Example \(\PageIndex{8}\)

    \(\dfrac{3 + \sqrt{8}}{3 - \sqrt{8}}\) We'll rationalize the denominator by multiplying this fraction by \(1\) in the form \(\dfrac{3 + \sqrt{8}}{3 + \sqrt{8}}\).

    \(\begin{array}{flushleft}
    \dfrac{3+\sqrt{8}}{3-\sqrt{8}} \cdot \frac{3+\sqrt{8}}{3+\sqrt{8}} &=\dfrac{(3+\sqrt{8})(3+\sqrt{8})}{3^{2}-(\sqrt{8})^{2}} \\
    &=\dfrac{9+3 \sqrt{8}+3 \sqrt{8}+\sqrt{8} \sqrt{8}}{9-8} \\
    &=\dfrac{9+6 \sqrt{8}+8}{1} \\
    &=17+6 \sqrt{8} \\
    &=17+6 \sqrt{4 \cdot 2} \\
    &=17+12 \sqrt{2}
    \end{array}\)

    Example \(\PageIndex{9}\)

    \(\dfrac{2 + \sqrt{7}}{4 - \sqrt{3}}\). Rationalize the denominator by multiplying this fraction by \(1\) in the form \(\dfrac{4 + \sqrt{3}}{4 + \sqrt{3}}\).

    \(\begin{array}{flushleft}
    \dfrac{2+\sqrt{7}}{4-\sqrt{3}} \cdot \dfrac{4+\sqrt{3}}{4+\sqrt{3}} &=\dfrac{(2+\sqrt{7})(4+\sqrt{3})}{4^{2}-(\sqrt{3})^{2}} \\
    &=\dfrac{8+2 \sqrt{3}+4 \sqrt{7}+\sqrt{21}}{16-3} \\
    &=\dfrac{8+2 \sqrt{3}+4 \sqrt{7}+\sqrt{21}}{13}
    \end{array}\)

    Practice Set B

    Simplify each by performing the indicated operation.

    Practice Problem \(\PageIndex{7}\)

    \(\sqrt{5}(\sqrt{6} - 4)\)

    Answer

    \(\sqrt{30} - 4\sqrt{5}\)

    Practice Problem \(\PageIndex{8}\)

    \((\sqrt{5} + \sqrt{7})(\sqrt{2} + \sqrt{8})\)

    Answer

    \(3\sqrt{10} + 3\sqrt{14}\)

    Practice Problem \(\PageIndex{9}\)

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

    Answer

    \(8\sqrt{6} - 12\)

    Practice Problem \(\PageIndex{10}\)

    \(\dfrac{4 + \sqrt{5}}{3 - \sqrt{8}}\)

    Answer

    \(12 + 8\sqrt{2} + 3\sqrt{5} + 2\sqrt{10}\)

    Exercises

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

    Exercise \(\PageIndex{1}\)

    \(4\sqrt{5} - 2\sqrt{5}\)

    Answer

    \(2\sqrt{5}\)

    Exercise \(\PageIndex{2}\)

    \(10 \sqrt{2} + 8\sqrt{2}\)

    Exercise \(\PageIndex{3}\)

    \(-3\sqrt{6} - 12\sqrt{6}\)

    Answer

    \(-15 \sqrt{6}\)

    Exercise \(\PageIndex{4}\)

    \(-\sqrt{10} - 2\sqrt{10}\)

    Exercise \(\PageIndex{5}\)

    \(3\sqrt{7x} + 2\sqrt{7x}\)

    Answer

    \(5\sqrt{7x}\)

    Exercise \(\PageIndex{6}\)

    \(6\sqrt{3a} + \sqrt{3a}\)

    Exercise \(\PageIndex{7}\)

    \(2\sqrt{18} + 5\sqrt{32}\)

    Answer

    \(26\sqrt{2}\)

    Exercise \(\PageIndex{8}\)

    \(4\sqrt{27} - 3\sqrt{48}\)

    Exercise \(\PageIndex{9}\)

    \(\sqrt{200} - \sqrt{128}\)

    Answer

    \(2\sqrt{2}\)

    Exercise \(\PageIndex{10}\)

    \(4\sqrt{300} + 2\sqrt{500}\)

    Exercise \(\PageIndex{11}\)

    \(6\sqrt{40} + 8\sqrt{80}\)

    Answer

    \(12\sqrt{10} + 32\sqrt{5}\)

    Exercise \(\PageIndex{12}\)

    \(2\sqrt{120} - 5\sqrt{30}\)

    Exercise \(\PageIndex{13}\)

    \(8\sqrt{60} - 3\sqrt{15}\)

    Answer

    \(13\sqrt{15}\)

    Exercise \(\PageIndex{14}\)

    \(\sqrt{a^3} - 3a\sqrt{a}\)

    Exercise \(\PageIndex{15}\)

    \(\sqrt{4x^3} + x\sqrt{x}\)

    Answer

    \(3x\sqrt{x}\)

    Exercise \(\PageIndex{16}\)

    \(2b\sqrt{a^3b^5} + 6a\sqrt{ab^7}\)

    Exercise \(\PageIndex{17}\)

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

    Answer

    \(2xy^2\sqrt{2xy}\)

    Exercise \(\PageIndex{18}\)

    \(5\sqrt{20} + 3\sqrt{45} - 3\sqrt{40}\)

    Exercise \(\PageIndex{19}\)

    \(\sqrt{24} - 2\sqrt{54} - 4\sqrt{12}\)

    Answer

    \(-4\sqrt{6} - 8\sqrt{3}\)

    Exercise \(\PageIndex{20}\)

    \(6\sqrt{18} + 5\sqrt{32} + 4\sqrt{50}\)

    Exercise \(\PageIndex{21}\)

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

    Answer

    \(-\sqrt{5}\)

    Exercise \(\PageIndex{22}\)

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

    Exercise \(\PageIndex{23}\)

    \(\sqrt{14} + 2\sqrt{56} - 3\sqrt{136}\)

    Answer

    \(5\sqrt{14} - 6\sqrt{34}\)

    Exercise \(\PageIndex{24}\)

    \(3\sqrt{2} + 2\sqrt{63} + 5\sqrt{7}\)

    Exercise \(\PageIndex{25}\)

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

    Answer

    \(13ax\sqrt{3x}\)

    Exercise \(\PageIndex{26}\)

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

    Exercise \(\PageIndex{27}\)

    \(\sqrt{2}(\sqrt{3} + 1)\)

    Answer

    \(\sqrt{6} + \sqrt{2}\)

    Exercise \(\PageIndex{28}\)

    \(\sqrt{3}(\sqrt{5} - 3)\)

    Exercise \(\PageIndex{29}\)

    \(\sqrt{5}(\sqrt{3} - \sqrt{2})\)

    Answer

    \(\sqrt{15} - \sqrt{10}\)

    Exercise \(\PageIndex{30}\)

    \(\sqrt{7}(\sqrt{6} - \sqrt{3})\)

    Exercise \(\PageIndex{31}\)

    \(\sqrt{8}(\sqrt{3} + \sqrt{2})\)

    Answer

    \(2(\sqrt{6} + 2)\)

    Exercise \(\PageIndex{32}\)

    \(\sqrt{10}(\sqrt{10} - \sqrt{5})\)

    Exercise \(\PageIndex{33}\)

    \((1 + \sqrt{3})(2 - \sqrt{3})\)

    Answer

    \(-1 + \sqrt{3}\)

    Exercise \(\PageIndex{34}\)

    \((5 + \sqrt{6})(4 - \sqrt{6})\)

    Exercise \(\PageIndex{35}\)

    \((3 - \sqrt{2})(4 - \sqrt{2})\)

    Answer

    \(7(2 - \sqrt{2})\)

    Exercise \(\PageIndex{36}\)

    \((5 + \sqrt{7})(4 - \sqrt{7})\)

    Exercise \(\PageIndex{37}\)

    \((\sqrt{2} + \sqrt{5})(\sqrt{2} + 3\sqrt{5})\)

    Answer

    \(17 + 4\sqrt{10}\)

    Exercise \(\PageIndex{38}\)

    \((2\sqrt{6} - \sqrt{3})(3\sqrt{6} + 2\sqrt{3})\)

    Exercise \(\PageIndex{39}\)

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

    Answer

    \(54 - 2\sqrt{15}\)

    Exercise \(\PageIndex{40}\)

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

    Exercise \(\PageIndex{41}\)

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

    Answer

    \(-42\)

    Exercise \(\PageIndex{42}\)

    \((1 + \sqrt{3})^2\)

    Exercise \(\PageIndex{43}\)

    \((3 + \sqrt{5})^2\)

    Answer

    \(14 + 6\sqrt{5}\)

    Exercise \(\PageIndex{44}\)

    \((2 - \sqrt{6})^2\)

    Exercise \(\PageIndex{45}\)

    \((2 - \sqrt{7})^2\)

    Answer

    \(11 - 4\sqrt{7}\)

    Exercise \(\PageIndex{46}\)

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

    Exercise \(\PageIndex{47}\)

    \((2 + \sqrt{5x})^2\)

    Answer

    \(4 + 4\sqrt{5x} + 5x\)

    Exercise \(\PageIndex{48}\)

    \((3 - \sqrt{3x})^2\)

    Exercise \(\PageIndex{49}\)

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

    Answer

    \(64 - 16\sqrt{6b} + 6b\)

    Exercise \(\PageIndex{50}\)

    \((2a + \sqrt{5a})^2\)

    Exercise \(\PageIndex{51}\)

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

    Answer

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

    Exercise \(\PageIndex{52}\)

    \((3 + \sqrt{3})(3 - \sqrt{3})\)

    Exercise \(\PageIndex{53}\)

    \((2 + \sqrt{5})(2 - \sqrt{5})\)

    Answer

    \(-1\)

    Exercise \(\PageIndex{54}\)

    \((8 + \sqrt{10})(8 - \sqrt{10})\)

    Exercise \(\PageIndex{55}\)

    \((6 + \sqrt{7})(6 - \sqrt{7})\)

    Answer

    \(29\)

    Exercise \(\PageIndex{56}\)

    \((\sqrt{2} + \sqrt{3})(\sqrt{2} - \sqrt{3}\)

    Exercise \(\PageIndex{57}\)

    \((\sqrt{5} + \sqrt{2})(\sqrt{5} - \sqrt{2})\)

    Answer

    \(3\)

    Exercise \(\PageIndex{58}\)

    \((\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b})\)

    Exercise \(\PageIndex{59}\)

    \((\sqrt{x} + \sqrt{y})(\sqrt{x} - \sqrt{y})\)

    Answer

    \(x - y\)

    Exercise \(\PageIndex{60}\)

    \(\dfrac{2}{5 + \sqrt{3}}\)

    Exercise \(\PageIndex{61}\)

    \(\dfrac{4}{6 + \sqrt{2}}\)

    Answer

    \(\dfrac{2(6 - \sqrt{2})}{17}\)

    Exercise \(\PageIndex{62}\)

    \(\dfrac{1}{3 - \sqrt{2}}\)

    Exercise \(\PageIndex{63}\)

    \(\dfrac{1}{4 - \sqrt{3}}\)

    Answer

    \(\dfrac{4 + \sqrt{3}}{13}\)

    Exercise \(\PageIndex{64}\)

    \(\dfrac{8}{2 - \sqrt{6}}\)

    Exercise \(\PageIndex{65}\)

    \(\dfrac{2}{3 - \sqrt{7}}\)

    Answer

    \(3 + \sqrt{7}\)

    Exercise \(\PageIndex{66}\)

    \(\dfrac{\sqrt{5}}{3 + \sqrt{3}}\)

    Exercise \(\PageIndex{67}\)

    \(\dfrac{\sqrt{3}}{6 + \sqrt{6}}\)

    Answer

    \(\dfrac{2\sqrt{3} - \sqrt{2}}{10}\)

    Exercise \(\PageIndex{68}\)

    \(\dfrac{2 - \sqrt{8}}{2 + \sqrt{8}}\)

    Exercise \(\PageIndex{69}\)

    \(\dfrac{4 + \sqrt{5}}{4 - \sqrt{5}}\)

    Answer

    \(\dfrac{21 + 8\sqrt{5}}{11}\)

    Exercise \(\PageIndex{70}\)

    \(\dfrac{1 + \sqrt{6}}{1 - \sqrt{6}}\)

    Exercise \(\PageIndex{71}\)

    \(\dfrac{8 - \sqrt{3}}{2 + \sqrt{18}}\)

    Answer

    \(\dfrac{-16 + 2\sqrt{3} + 24\sqrt{2} - 3\sqrt{6}}{14}\)

    Exercise \(\PageIndex{72}\)

    \(\dfrac{6 - \sqrt{2}}{4 + \sqrt{12}}\)

    Exercise \(\PageIndex{73}\)

    \(\dfrac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}}\)

    Answer

    \(5 - 2\sqrt{6}\)

    Exercise \(\PageIndex{74}\)

    \(\dfrac{\sqrt{6a} - \sqrt{8a}}{\sqrt{8a} + \sqrt{6a}}\)

    Exercise \(\PageIndex{75}\)

    \(\dfrac{\sqrt{2b} - \sqrt{3b}}{\sqrt{3b} + \sqrt{2b}}\)

    Answer

    \(2\sqrt{6} - 5\)

    Exercises For Review

    Exercise \(\PageIndex{76}\)

    Simplify \((\dfrac{x^5y^3}{x^2y})^5\)

    Exercise \(\PageIndex{77}\)

    Simplify \((8x^3y)^2(x^2y^3)^4\)

    Answer

    \(64x^{14}y^{14}\)

    Exercise \(\PageIndex{78}\)

    Write \((x-1)^4(x-1)^{-7}\) so that only positive exponents appear.

    Exercise \(\PageIndex{79}\)

    Simpify \(\sqrt{27x^5y^{10}z^3}\)

    Answer

    \(3x^2y^5z\sqrt{3xz}\)

    Exercise \(\PageIndex{80}\)

    Simplify \(\dfrac{1}{2 + \sqrt{x}}\) by rationalizing the denominator.


    This page titled 9.6: Addition and Subtraction of Square Root Expressions is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Denny Burzynski & Wade Ellis, Jr. (OpenStax CNX) .

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