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Mathematics LibreTexts

9.4: Multiply Square Roots

  • Page ID
    18990
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    By the end of this section, you will be able to:
    • Multiply square roots
    • Use polynomial multiplication to multiply square roots

    Before you get started, take this readiness quiz.

    1. Simplify: (3u)(8v).
      If you missed this problem, review [link].
    2. Simplify: 6(12−7n).
      If you missed this problem, review [link].
    3. Simplify: (2+a)(4−a).
      If you missed this problem, review [link].

    Multiply Square Roots

    We have used the Product Property of Square Roots to simplify square roots by removing the perfect square factors. The Product Property of Square Roots says

    \(\sqrt{ab}=\sqrt{a}·\sqrt{b}\)

    We can use the Product Property of Square Roots ‘in reverse’ to multiply square roots.

    \(\sqrt{a}·\sqrt{b}=\sqrt{ab}\)

    Remember, we assume all variables are greater than or equal to zero.

    We will rewrite the Product Property of Square Roots so we see both ways together.

    Definition: PRODUCT PROPERTY OF SQUARE ROOTS

    If a, b are nonnegative real numbers, then

    \(\sqrt{ab}=\sqrt{a}·\sqrt{b}\) and \(\sqrt{a}·\sqrt{b}=\sqrt{ab}\).

    So we can multiply \(\sqrt{3}·\sqrt{5}\) in this way:

    \[\begin{array}{l} {\sqrt{3}·\sqrt{5}}\\ {\sqrt{3·5}}\\ {\sqrt{15}}\\ \nonumber \end{array}\]

    Sometimes the product gives us a perfect square:

    \[\begin{array}{l} {\sqrt{2}·\sqrt{8}}\\ {\sqrt{2·8}}\\ {\sqrt{16}}\\ {4}\\ \nonumber \end{array}\]

    Even when the product is not a perfect square, we must look for perfect-square factors and simplify the radical whenever possible.

    Multiplying radicals with coefficients is much like multiplying variables with coefficients. To multiply 4x·3y we multiply the coefficients together and then the variables. The result is 12xy. Keep this in mind as you do these examples.

    Example \(\PageIndex{1}\)

    Simplify:

    1. \(\sqrt{2}·\sqrt{6}\)
    2. \((4\sqrt{3})(2\sqrt{12})\).
    Answer
    1. \(\sqrt{2}·\sqrt{6}\)
    Multiply using the Product Property. \(\sqrt{12}\)
    Simplify the radical. \(\sqrt{4}·\sqrt{3}\)
    Simplify. \(2\sqrt{3}\)
    2. \((4\sqrt{3})(2\sqrt{12})\)
    Multiply using the Product Property. \(8\sqrt{36}\)
    Simplify the radical. \(8·6\)
    Simplify. \(48\)

    Example \(\PageIndex{2}\)

    Simplify:

    1. \(\sqrt{3}·\sqrt{6}\)
    2. \((2\sqrt{6})(3\sqrt{12})\).
    Answer
    1. \(3\sqrt{2}\)
    2. \(36\sqrt{2}\)

    Example \(\PageIndex{3}\)

    Simplify:

    1. \(\sqrt{5}·\sqrt{10}\)
    2. \((6\sqrt{3})(5\sqrt{6})\)
    Answer
    1. \(5\sqrt{2}\)
    2. \(90\sqrt{2}\)

    Example \(\PageIndex{4}\)

    Simplify: \((6\sqrt{2})(3\sqrt{10})\)

    Answer
    \((6\sqrt{2})(3\sqrt{10})\)
    Multiply using the Product Property. \(18\sqrt{20}\)
    Simplify the radical. \(18\sqrt{4}·\sqrt{5}\)
    Simplify. \(18·2·\sqrt{5}\)
    \(36\sqrt{5}\)

    Example \(\PageIndex{5}\)

    Simplify: \((3\sqrt{2})(2\sqrt{30})\)

    Answer

    \(12\sqrt{15}\)

    Example \(\PageIndex{6}\)

    Simplify: \((3\sqrt{3})(3\sqrt{6})\).

    Answer

    \(27\sqrt{2}\)

    When we have to multiply square roots, we first find the product and then remove any perfect square factors.

    Example \(\PageIndex{7}\)

    Simplify:

    1. \((\sqrt{8x^3})(\sqrt{3x})\)
    2. \((\sqrt{20y^2})(\sqrt{5y^3})\)
    Answer
    1. \((\sqrt{8x^3})(\sqrt{3x})\)
    Multiply using the Product Property. \(\sqrt{24x^4}\)
    Simplify the radical. \(\sqrt{4x^4}·\sqrt{6}\)
    Simplify. \(2x^2\sqrt{6}\)
    2. \((\sqrt{20y^2})(\sqrt{5y^3})\)
    Multiply using the Product Property. \(\sqrt{100y^5}\)
    Simplify the radical. \(10y^2\sqrt{y}\)

    Example \(\PageIndex{8}\)

    Simplify:

    1. \((\sqrt{6x^3})(\sqrt{3x})\)
    2. \((\sqrt{2y^3})(\sqrt{50y^2})\).
    Answer
    1. \(3x^2\sqrt{2}\)
    2. \(10y^2\sqrt{y}\)

    Example \(\PageIndex{9}\)

    Simplify:

    1. \((\sqrt{6x^5})(\sqrt{2x})\)
    2. \((\sqrt{12y^2})(\sqrt{3y^5})\)
    Answer
    1. \(2x^3\sqrt{3}\)
    2. \(6y^2\sqrt{y}\)

    Example \(\PageIndex{10}\)

    Simplify: \((10\sqrt{6p^3})(3\sqrt{18p})\)

    Answer
    \((10\sqrt{6p^3})(3\sqrt{18p})\)
    Multiply. \(30\sqrt{108p^4}\)
    Simplify the radical. \(30\sqrt{36p^4}·\sqrt{3}\)
    \(30·6p^2·\sqrt{3}\)
    \(180p^2\sqrt{3}\)

    Example \(\PageIndex{11}\)

    Simplify: \((\sqrt{62x^2})(8\sqrt{45x^4})\)

    Answer

    \(144x^3\sqrt{10}\)

    Example \(\PageIndex{12}\)

    Simplify: \((2\sqrt{6y^4})(12\sqrt{30y})\).

    Answer

    \(144y^2\sqrt{5y}\)

    Example \(\PageIndex{13}\)

    Simplify:

    1. \((\sqrt{2})^2\)
    2. \((−\sqrt{11})^2\).
    Answer
    1. \((\sqrt{2})^2\)
    Rewrite as a product. \((\sqrt{2})(\sqrt{2})\)
    Multiply. \(\sqrt{4}\)
    Simplify. 2
    2. \((−\sqrt{11})^2\)
    Rewrite as a product. \((−\sqrt{11})(−\sqrt{11})\)
    Multiply. \(\sqrt{121}\)
    Simplify. 11

    Example \(\PageIndex{14}\)

    Simplify:

    1. \((\sqrt{12})^2\)
    2. \((−\sqrt{15})^2\).
    Answer
    1. 12
    2. 15

    Example \(\PageIndex{15}\)

    Simplify:

    1. \((\sqrt{16})^2\)
    2. \((−\sqrt{20})^2\).
    Answer
    1. 16
    2. 20

    The results of the previous example lead us to this property.

    Definition: SQUARING A SQUARE ROOT

    If a is a nonnegative real number, then

    \((\sqrt{a})^2=a\)

    By realizing that squaring and taking a square root are ‘opposite’ operations, we can simplify \((\sqrt{2})^2\) and get 2 right away. When we multiply the two like square roots in part (a) of the next example, it is the same as squaring.

    Example \(\PageIndex{16}\)

    Simplify:

    1. \((2\sqrt{3})(8\sqrt{3})\)
    2. \((3\sqrt{6})^2\).
    Answer
    1. \((2\sqrt{3})(8\sqrt{3})\)
    Multiply. Remember, \((\sqrt{3}^2)\) 16·3
    Simplify. 48
    2. \((3\sqrt{6})^2\)
    Multiply. 9·6
    Simplify. 54

    Example \(\PageIndex{17}\)

    Simplify:

    1. \((6\sqrt{11})(5\sqrt{11})\)
    2. \((5\sqrt{8})^2\).
    Answer
    1. 330
    2. 200

    Example \(\PageIndex{18}\)

    Simplify:

    1. \((3\sqrt{7})(10\sqrt{7})\)
    2. \((−4\sqrt{6})^2\).
    Answer
    1. 210
    2. 96

    Use Polynomial Multiplication to Multiply Square Roots

    In the next few examples, we will use the Distributive Property to multiply expressions with square roots.

    We will first distribute and then simplify the square roots when possible.

    Example \(\PageIndex{19}\)

    Simplify:

    1. \(3(5−\sqrt{2})\)
    2. \(\sqrt{2}(4−\sqrt{10})\).
    Answer
    1. \(3(5−\sqrt{2})\)
    Distribute. \(15−3\sqrt{2})\)
    2. \(\sqrt{2}(4−\sqrt{10})\)
    Distribute. \(4\sqrt{2}−\sqrt{20}\)
    Simplify. \(4\sqrt{2}−2\sqrt{5}\)

    Example \(\PageIndex{20}\)

    Simplify:

    1. \(2(3−\sqrt{5})\)
    2. \(\sqrt{3}(2−\sqrt{18})\).
    Answer
    1. \(6−2\sqrt{5}\)
    2. \(2\sqrt{3}−3\sqrt{6}\)

    Example \(\PageIndex{21}\)

    Simplify:

    1. \(6(2+\sqrt{6})\)
    2. \(\sqrt{7}(1+\sqrt{14})\).
    Answer
    1. \(12+\sqrt{6}\)
    2. \(\sqrt{7}+7\sqrt{2}\)

    Example \(\PageIndex{22}\)

    Simplify:

    1. \(\sqrt{5}(7+2\sqrt{5})\)
    2. \(\sqrt{6}(\sqrt{2}+\sqrt{18})\).
    Answer
    1. \(\sqrt{5}(7+2\sqrt{5})\)
    Distribute. \(7\sqrt{5}+2·5\)
    Simplify. \(7\sqrt{5}+10\)
    \(10+7\sqrt{5}\)
    2. \(\sqrt{6}(\sqrt{2}+\sqrt{18})\)
    Multiply. \(\sqrt{12}+\sqrt{108}\)
    Simplify. \((\sqrt{4}·\sqrt{3}+\sqrt{36}·\sqrt{3})\)
    \(2\sqrt{3}+6\sqrt{3}\)
    Combine like radicals. \(8\sqrt{3}\)

    Example \(\PageIndex{23}\)

    Simplify:

    1. \(\sqrt{6}(1+3\sqrt{6})\)
    2. \(\sqrt{12}(\sqrt{3}+\sqrt{24})\)
    Answer
    1. \(18+\sqrt{6}\)
    2. \(6+12\sqrt{2}\)

    Example \(\PageIndex{24}\)

    Simplify:

    1. \(\sqrt{8}(2−5\sqrt{8})\)
    2. \(\sqrt{14}(\sqrt{2}+\sqrt{42})\)
    Answer
    1. \(−40+4\sqrt{2}\)
    2. \(2\sqrt{7}+14\sqrt{3}\)

    When we worked with polynomials, we multiplied binomials by binomials. Remember, this gave us four products before we combined any like terms. To be sure to get all four products, we organized our work—usually by the FOIL method.

    Example \(\PageIndex{25}\)

    Simplify: \((2+\sqrt{3})(4−\sqrt{3})\)

    Answer
    \((2+\sqrt{3})(4−\sqrt{3})\)
    Multiply. \(8−2\sqrt{3}+4\sqrt{3}−3\)
    Combine like terms. \(5+2\sqrt{3}\)

    Example \(\PageIndex{26}\)

    Simplify: \((1+\sqrt{6})(3−\sqrt{6})\).

    Answer

    \(−3+2\sqrt{6}\)

    Example \(\PageIndex{27}\)

    Simplify: \((4−\sqrt{10})(2+\sqrt{10})\).

    Answer

    \(−2+2\sqrt{10}\)

    Example \(\PageIndex{28}\)

    Simplify: \((3−2\sqrt{7})(4−2\sqrt{7})\).

    Answer
    \((3−2\sqrt{7})(4−2\sqrt{7})\)
    Multiply. \(12−6\sqrt{7}−8\sqrt{7}+4·7\)
    Simplify. \(12−6\sqrt{7}−8\sqrt{7}+28\)
    Combine like terms. \(40−14\sqrt{7}\)

    Example \(\PageIndex{29}\)

    Simplify: \((6−3\sqrt{7})(3+4\sqrt{7})\).

    Answer

    \(−66+15\sqrt{7}\)

    Example \(\PageIndex{30}\)

    Simplify: \((2−3\sqrt{11})(4−\sqrt{11})\)

    Answer

    \(41+14\sqrt{11}\)

    Example \(\PageIndex{31}\)

    Simplify: \((3\sqrt{2}−\sqrt{5})(\sqrt{2}+4\sqrt{5})\).

    Answer
    \(3\sqrt{2}−\sqrt{5})(\sqrt{2}+4\sqrt{5})\)
    Multiply. \(3·2+12\sqrt{10}−\sqrt{10}−4·5\)
    Simplify. \(6+12\sqrt{10}−\sqrt{10}−20\)
    Combine like terms. \(−14+11\sqrt{10}\)

    Example \(\PageIndex{32}\)

    Simplify: \((5\sqrt{3}−\sqrt{7})(\sqrt{3}+2\sqrt{7})\)

    Answer

    \(1+9\sqrt{21}\)

    Example \(\PageIndex{33}\)

    Simplify: \((\sqrt{6}−3\sqrt{8})(2\sqrt{6}+\sqrt{8})\)

    Answer

    \(−12−20\sqrt{3}\)

    Example \(\PageIndex{34}\)

    Simplify: \((4−2\sqrt{x})(1+3\sqrt{x})\).

    Answer
    \((4−2\sqrt{x})(1+3\sqrt{x})\).
    Multiply. \(4+12\sqrt{x}−2\sqrt{x}−6x\)
    Combine like terms. \(4+10\sqrt{x}−6x\)

    Example \(\PageIndex{35}\)

    Simplify: \((6−5\sqrt{m})(2+3\sqrt{m})\).

    Answer

    \(12−8\sqrt{m}−15m\)

    Example \(\PageIndex{36}\)

    Simplify: \((10+3\sqrt{n})(1−5\sqrt{n})\)

    Answer

    \(10−47\sqrt{n}−15n\)

    Note that some special products made our work easier when we multiplied binomials earlier. This is true when we multiply square roots, too. The special product formulas we used are shown below.

    Definition: SPECIAL PRODUCT FORMULAS

    \[\begin{array}{ll} {\textbf{Binomial Squares}}&{\textbf{Product of Conjugates}}\\ {(a+b)^2=a^2+2ab+b^2}&{(a−b)(a+b)=a^2−b^2}\\ {(a−b)^2=a^2−2ab+b^2}&{}\\ \nonumber \end{array}\]

    ​​​​​​​We will use the special product formulas in the next few examples. We will start with the Binomial Squares formula.​​​​​​​

    Example \(\PageIndex{37}\)

    Simplify:

    1. \((2+\sqrt{3})^2\)
    2. \((4−2\sqrt{5})^2\).
    Answer

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

    1.

    .
    Multiply using the binomial square pattern. .
    Simplify. .
    Combine like terms. .
    2.
    .
    Multiply using the binomial square pattern. .
    Simplify. .
    Combine like terms. .

    Example \(\PageIndex{38}\)

    Simplify:

    1. \((10+\sqrt{2})^2\)
    2. \((1+3\sqrt{6})^2\).
    Answer
    1. \(102+20\sqrt{2}\)
    2. \(55+6\sqrt{6}\)

    Example \(\PageIndex{39}\)

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

    Answer
    .
    Multiply using the binomial square pattern. .
    Simplify. .

    Example \(\PageIndex{40}\)

    Simplify: \((2+5\sqrt{m})^2\).

    Answer

    \(4+20\sqrt{m}+25m\)

    Example \(\PageIndex{41}\)

    Simplify: \((3−4\sqrt{n})^2\).

    Answer

    \(9−24\sqrt{n}+16n\)

    ​​​​​​​In the next two examples, we will find the product of conjugates.

    Example \(\PageIndex{42}\)

    Simplify: \((4−\sqrt{2})(4+\sqrt{2})\).

    Answer
    .
    Multiply using the binomial square pattern. .
    Simplify. .

    Example \(\PageIndex{43}\)

    Simplify: \((2−\sqrt{3})(2+\sqrt{3})\)

    Answer

    1

    ExAMPLe \(\PageIndex{44}\)

    Simplify: \((1+\sqrt{5})(1−\sqrt{5})\)

    Answer

    −4

    Example \(\PageIndex{45}\)

    Simplify: \((5−2\sqrt{3})(5+2\sqrt{3})\)

    Answer
    .
    Multiply using the binomial square pattern. .
    Simplify. .

    Example \(\PageIndex{46}\)

    Simplify: \((3−2\sqrt{5})(3+2\sqrt{5})\).

    Answer

    −11

    Example \(\PageIndex{47}\)

    Simplify: \((4+5\sqrt{7})(4−5\sqrt{7})\).

    Answer

    −159

    ​​​​​​​Access these online resources for additional instruction and practice with multiplying square roots.

    Key Concepts

    • Product Property of Square Roots If a, b are nonnegative real numbers, then

      \(\sqrt{ab}=\sqrt{a·b}\) and \(\sqrt{a}·\sqrt{b}=\sqrt{ab}\)

    • Special formulas for multiplying binomials and conjugates:

      \[\begin{array}{ll} {(a+b)^2=a^2+2ab+b^2}&{(a−b)(a+b)=a^2−b^2}\\ {(a−b)^2=a^2−2ab+b^2 }&{}\\ \nonumber \end{array}\]

    • The FOIL method can be used to multiply binomials containing radicals.

    Practice Makes Perfect

    Multiply Square Roots

    In the following exercises, simplify.

    Example \(\PageIndex{48}\)

    1. \(\sqrt{2}·\sqrt{8}\)
    2. \((3\sqrt{3})(2\sqrt{18})\)
    Answer
    1. 44
    2. 36

    Example \(\PageIndex{49}\)

    1. \(\sqrt{6}·\sqrt{6}\)
    2. \((3\sqrt{2})(2\sqrt{32})\)

    Example \(\PageIndex{50}\)

    1. \(\sqrt{7}·\sqrt{14}\)
    2. \((4\sqrt{8})(5\sqrt{8})\)
    Answer
    1. \(7\sqrt{2}\)
    2. 160

    Example \(\PageIndex{51}\)

    1. \(\sqrt{6}·\sqrt{12}\)
    2. \((2\sqrt{5})(2\sqrt{10})\)

    Example \(\PageIndex{52}\)

    \((5\sqrt{2})(3\sqrt{6})\)

    Answer

    \(30\sqrt{3}\)

    Example \(\PageIndex{53}\)

    \((2\sqrt{3})(4\sqrt{6})\)

    Example \(\PageIndex{54}\)

    \((−2\sqrt{3})(3\sqrt{18})\)

    Answer

    \(−18\sqrt{6}\)

    Example \(\PageIndex{55}\)

    \((−4\sqrt{5})(5\sqrt{10})\)

    Example \(\PageIndex{56}\)

    \((5\sqrt{6})(−\sqrt{12})\)

    Answer

    \(−30\sqrt{2}\)

    Example \(\PageIndex{57}\)

    \((6\sqrt{2})(−\sqrt{10})\)

    Example \(\PageIndex{58}\)

    \((−2\sqrt{7})(−2\sqrt{14})\)

    Answer

    \(28\sqrt{2}\)

    Example \(\PageIndex{59}\)

    \((−2\sqrt{11})(−4\sqrt{22})\)

    Example \(\PageIndex{60}\)

    1. \((\sqrt{15y})(\sqrt{5y^3})\)
    2. \((\sqrt{2n^2})(\sqrt{18n^3})\)
    Answer
    1. \(5y^2\sqrt{3}\)
    2. \(6n^2\sqrt{n}\)

    Example \(\PageIndex{61}\)

    1. \((\sqrt{14x^3})(\sqrt{7x^3})\)
    2. \((\sqrt{3q^2})(\sqrt{48q^3})\)

    Example \(\PageIndex{62}\)

    1. \((\sqrt{16y^2})(\sqrt{8y^4})\)
    2. \((\sqrt{11s^6})(\sqrt{11s})\)
    Answer
    1. \(8y^3\sqrt{2}\)
    2. \(11s^3\sqrt{s}\)

    Example \(\PageIndex{63}\)

    ⓐ \((\sqrt{8x^3})(\sqrt{3x})\)
    ⓑ \((\sqrt{7r})(\sqrt{7r^8})\)

    Example \(\PageIndex{64}\)

    \((2\sqrt{5b^3})(4\sqrt{15b})\)

    Answer

    \(40b^2\sqrt{3}\)

    Example \(\PageIndex{65}\)

    \((\sqrt{38c^5})(\sqrt{26c^3})\)​​​​​​​

    Example \(\PageIndex{66}\)

    \((6\sqrt{3d^3})(4\sqrt{12d^5})\)

    Answer

    \(144d^4\)​​​​​​​

    Example \(\PageIndex{67}\)

    \((2\sqrt{5b^3})(4\sqrt{15b})\)

    Example \(\PageIndex{68}\)

    \((6\sqrt{3d^3})(4\sqrt{12d^5})\)

    Answer

    \(54y^4\sqrt{y}\)

    Example \(\PageIndex{69}\)

    \((−2\sqrt{7z^3})(3\sqrt{14z^8})\)

    Example \(\PageIndex{70}\)

    \((4\sqrt{2k^5})(−3\sqrt{32k^6})\)

    Answer

    \(−96k^5\sqrt{k}\)

    Example \(\PageIndex{71}\)

    1. \((\sqrt{7})^2\)
    2. \((−\sqrt{15})^2\)

    Example \(\PageIndex{72}\)

    1. \((\sqrt{11})^2\)
    2. \((−\sqrt{21})^2\)
    Answer
    1. 11
    2. 21

    Example \(\PageIndex{73}\)

    1. \((\sqrt{19})^2\)
    2. \((−\sqrt{5})^2\)

    Exercise \(\PageIndex{74}\)

    1. \((\sqrt{23})^2\)
    2. \((−\sqrt{3})^2\)
    Answer
    1. 23
    2. 3

    Example \(\PageIndex{75}\)

    1. \((4\sqrt{11})(−3\sqrt{11})\)
    2. \((5\sqrt{3})^2\)

    Example \(\PageIndex{76}\)

    1. \((2\sqrt{13})(−9\sqrt{13})\)
    2. \((6\sqrt{5})^2\)
    Answer
    1. −234
    2. 180

    Example \(\PageIndex{77}\)

    1. \((−3\sqrt{12})(−2\sqrt{6})\)
    2. \( (−4\sqrt{10})^2\)

    Example \(\PageIndex{78}\)

    1. \((−7\sqrt{5})(−3\sqrt{10})\)
    2. \( (−2\sqrt{14})^2\)
    Answer
    1. \(105\sqrt{2}\)
    2. 56

    Use Polynomial Multiplication to Multiply Square Roots

    In the following exercises, simplify.

    Example \(\PageIndex{79}\)

    1. \(3(4−\sqrt{3})\)
    2. \(\sqrt{2}(4−\sqrt{6})\)

    Example \(\PageIndex{80}\)

    1. \(4(6−\sqrt{11})\)
    2. \(\sqrt{2}(5−\sqrt{12})\)
    Answer
    1. \(24−4\sqrt{11}\)
    2. \(5\sqrt{2}−4\sqrt{6}\)

    Example \(\PageIndex{81}\)

    1. \(5(3−\sqrt{7})\)
    2. \(\sqrt{3}(4−\sqrt{15})\)

    Example \(\PageIndex{82}\)

    1. \(7(−2−\sqrt{11})\)
    2. \(\sqrt{7}(6−\sqrt{14})\)
    Answer
    1. \(−14−7\sqrt{11}\)
    2. \(6\sqrt{7}−7\sqrt{2}\)

    Example \(\PageIndex{83}\)

    1. \(\sqrt{7}(5+2\sqrt{7})\)
    2. \(\sqrt{5}(\sqrt{10}+\sqrt{18})\)

    Example \(\PageIndex{84}\)

    1. \(\sqrt{11}(8+4\sqrt{11})\)
    2. \(\sqrt{3}(\sqrt{12}+\sqrt{27})\)
    Answer
    1. \(44+8\sqrt{11}\)
    2. 15

    Example \(\PageIndex{85}\)

    1. \(\sqrt{11}(−3+4\sqrt{1})\)
    2. \(\sqrt{3}(\sqrt{15}−\sqrt{18})\)

    Example \(\PageIndex{86}\)

    1. \(\sqrt{2}(−5+9\sqrt{2})\)
    2. \(\sqrt{7}(\sqrt{3}−\sqrt{21})\)
    Answer
    1. \(18−5\sqrt{2}\)
    2. \(\sqrt{21}−7\sqrt{3}\)

    Example \(\PageIndex{87}\)

    \((8+\sqrt{3})(2−\sqrt{3})\)

    Example \(\PageIndex{88}\)

    \((7+\sqrt{3})(9−\sqrt{3})\)

    Answer

    \(60+2\sqrt{3}\)

    Example \(\PageIndex{89}\)

    \((8−\sqrt{2})(3+\sqrt{2})\)

    Example \(\PageIndex{90}\)

    \((9−\sqrt{2})(6+\sqrt{2})\)

    Answer

    \(52+3\sqrt{2}\)

    Example \(\PageIndex{91}\)

    \((3−\sqrt{7})(5−\sqrt{7})\)

    Example \(\PageIndex{92}\)

    \((5−\sqrt{7})(4−\sqrt{7})\)

    Answer

    \(27−9\sqrt{7}\)

    Example \(\PageIndex{93}\)

    \((1+3\sqrt{10})(5−2\sqrt{10})\)

    Exercise \(\PageIndex{94}\)

    \((7−2\sqrt{5})(4+9\sqrt{5})\)

    Answer

    \(−62+55\sqrt{5}\)

    Example \(\PageIndex{95}\)

    \((\sqrt{3}+\sqrt{10})(\sqrt{3}+2\sqrt{10})\)

    Example \(\PageIndex{96}\)

    \((\sqrt{11}+\sqrt{5})(\sqrt{11}+6\sqrt{5})\)

    Answer

    \(161+7\sqrt{55}\)​​​​​​​

    Example \(\PageIndex{97}\)

    \((2\sqrt{7}−5\sqrt{11})(4\sqrt{7}+9\sqrt{11})\)

    Example \(\PageIndex{98}\)

    \((4\sqrt{6}+7\sqrt{13})(8\sqrt{6}−3\sqrt{13})\)

    Answer

    \(−81+44\sqrt{78}\)

    Example \(\PageIndex{99}\)

    \((5−\sqrt{u})(3+\sqrt{u})\)

    Example \(\PageIndex{100}\)

    \((9−\sqrt{w})(2+\sqrt{w})\)

    Answer

    \(18+7\sqrt{w}\)

    Example \(\PageIndex{101}\)

    \((7+2\sqrt{m})(4+9\sqrt{m})\)

    Example \(\PageIndex{102}\)

    \((6+5\sqrt{n})(11+3\sqrt{n})\)

    Answer

    \(66+73\sqrt{n}+15n\)

    Example \(\PageIndex{103}\)

    1. \((3+\sqrt{5})^2\)
    2. \((2−5\sqrt{3})^2\)

    Example \(\PageIndex{104}\)

    1. \((4+\sqrt{11})^2\)
    2. \((3−2\sqrt{5})^2\)
    Answer
    1. \(27+8\sqrt{11}\)
    2. \(29−12\sqrt{5}\)

    Example \(\PageIndex{105}\)

    1. \((9−\sqrt{6})^2\)
    2. \((10+3\sqrt{7})^2\)

    Example \(\PageIndex{106}\)

    1. \((5−\sqrt{10})^2\)
    2. \((8+3\sqrt{2})^2\)
    Answer
    1. \(35−10\sqrt{10}\)
    2. \(82+48\sqrt{2}\)

    Example \(\PageIndex{107}\)

    \((3−\sqrt{5})(3+\sqrt{5})\)

    Example \(\PageIndex{108}\)

    \((10−\sqrt{3})(10+\sqrt{3})\)

    Answer

    97

    Example \(\PageIndex{109}\)

    \((4+\sqrt{2})(4−\sqrt{2})\)​​​​​​​

    Example \(\PageIndex{110}\)

    \((7+\sqrt{10})(7−\sqrt{10})\)

    Answer

    39

    Example \(\PageIndex{111}\)

    \((4+9\sqrt{3})(4−9\sqrt{3})\)

    Example \(\PageIndex{112}\)

    \((1+8\sqrt{2})(1−8\sqrt{2})\)

    Answer

    −127

    Example \(\PageIndex{113}\)

    \((12−5\sqrt{5})(12+5\sqrt{5})\)

    Example \(\PageIndex{114}\)

    \((9−4\sqrt{3})(9+4\sqrt{3})\)

    Answer

    33

    Mixed Practice

    In the following exercises, simplify.

    Example \(\PageIndex{115}\)

    \(\sqrt{3}·\sqrt{21}\)

    Example \(\PageIndex{116}\)

    \((4\sqrt{6})(−\sqrt{18})\)

    Answer

    \(−24\sqrt{3}\)

    Example \(\PageIndex{117}\)

    \((−5+\sqrt{7})(6+\sqrt{21})\)

    Example \(\PageIndex{118}\)

    \((−5\sqrt{7})(6\sqrt{21})\)

    Answer

    \(−210\sqrt{3}\)

    Example \(\PageIndex{119}\)

    \((−4\sqrt{2})(2\sqrt{18})\)

    Example \(\PageIndex{120}\)

    \((\sqrt{35y^3})(\sqrt{7y^3})\)

    Answer

    \(7y^3\sqrt{5}\)

    Example \(\PageIndex{121}\)

    \((4\sqrt{12x^5})(2\sqrt{6x^3})\)

    Example \(\PageIndex{122}\)

    \((\sqrt{29})^2\)

    Answer

    29

    Example \(\PageIndex{123}\)

    \((−4\sqrt{17})(−3\sqrt{17})\)​​​​​​​

    Example \(\PageIndex{124}\)

    \((−4+\sqrt{17})(−3+\sqrt{17})\)

    Answer

    \(29−7\sqrt{17}\)

    Everyday Math

    Example \(\PageIndex{125}\)

    A landscaper wants to put a square reflecting pool next to a triangular deck, as shown below. The triangular deck is a right triangle, with legs of length 9 feet and 11 feet, and the pool will be adjacent to the hypotenuse.

    1. Use the Pythagorean Theorem to find the length of a side of the pool. Round your answer to the nearest tenth of a foot.
    2. Find the exact area of the pool.

    This figure is an illustration of a square pool with a deck in the shape of a right triangle. the pool's sides are x inches long while the deck's hypotenuse is x inches long and its legs are nine and eleven inches long.

    Example \(\PageIndex{126}\)

    An artist wants to make a small monument in the shape of a square base topped by a right triangle, as shown below. The square base will be adjacent to one leg of the triangle. The other leg of the triangle will measure 2 feet and the hypotenuse will be 5 feet.

    1. Use the Pythagorean Theorem to find the length of a side of the square base. Round your answer to the nearest tenth of a foot.
      This figure shows a marble sculpture in the form of a square with a right triangle resting on top of it. The sides of the square are x inches long, the legs of the triangle are x and two inches long, and the hypotenuse of the triangle is five inches long.
    2. Find the exact area of the face of the square base.
    Answer
    1. 4.6feet
    2. 21 sq. feet

    Example \(\PageIndex{127}\)

    A square garden will be made with a stone border on one edge. If only \(3+\sqrt{10}\) feet of stone are available, simplify \((3+\sqrt{10})^2\) to determine the area of the largest such garden.​​​​​​​

    Example \(\PageIndex{128}\)

    A garden will be made so as to contain two square sections, one section with side length \(\sqrt{5}+\sqrt{6}\) yards and one section with side length \(\sqrt{2}+\sqrt{3}\) yards. Simplify \((\sqrt{5}+\sqrt{6})(\sqrt{2}+\sqrt{3})\) to determine the total area of the garden.

    Example \(\PageIndex{129}\)

    Suppose a third section will be added to the garden in the previous exercise. The third section is to have a width of \(\sqrt{432}\) feet. Write an expression that gives the total area of the garden.​​

    Writing Exercises

    Example \(\PageIndex{130}\)

    1. Explain why \((−\sqrt{n})^2\) is always positive, for \(n \ge 0\).
    2. Explain why \(−(\sqrt{n})^2\) is always negative, for \(n \ge 0\).
    Answer
    1. when squaring a negative, it becomes a positive
    2. since the negative is not included in the parenthesis, it is not squared, and remains negative​​​​​​​

    Example \(\PageIndex{131}\)

    Use the binomial square pattern to simplify \((3+\sqrt{2})^2\). Explain all your steps.

    Self Check

    ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

    This table has four columns and three rows. The columns are labeled, “I can…,” “confidently.,” “with some help.,” and “no minus I don’t get it!” The rows under the “I can…” column read, “multiply square roots.,” and “use polynomial multiplication to multiply square roots.” The other rows under the other columns are empty.

    ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

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