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9.4E: Exercises

  • Page ID
    30278
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    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. \(18\sqrt{6}\)
    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 \(\PageIndex{68}\)

    \((2\sqrt{5d^6})(3\sqrt{20d^2})\)

    Answer

    \(60d^4\)

    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. \(5\sqrt{2}−2\sqrt{6}\)
    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

    \(41+7\sqrt{55}\)

    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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