13.8.4E: Exercises
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Practice Makes Perfect
Multiply Square Roots
In the following exercises, simplify.
Example \(\PageIndex{48}\)
- \(\sqrt{2}·\sqrt{8}\)
- \((3\sqrt{3})(2\sqrt{18})\)
- Answer
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- \(44\)
- \(18\sqrt{6}\)
Example \(\PageIndex{49}\)
- \(\sqrt{6}·\sqrt{6}\)
- \((3\sqrt{2})(2\sqrt{32})\)
Example \(\PageIndex{50}\)
- \(\sqrt{7}·\sqrt{14}\)
- \((4\sqrt{8})(5\sqrt{8})\)
- Answer
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- \(7\sqrt{2}\)
- 160
Example \(\PageIndex{51}\)
- \(\sqrt{6}·\sqrt{12}\)
- \((2\sqrt{5})(2\sqrt{10})\)
Example \(\PageIndex{52}\)
\((5\sqrt{2})(3\sqrt{6})\)
- Answer
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\(30\sqrt{3}\)
Example \(\PageIndex{53}\)
\((2\sqrt{3})(4\sqrt{6})\)
Example \(\PageIndex{54}\)
\((−2\sqrt{3})(3\sqrt{18})\)
- Answer
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\(−18\sqrt{6}\)
Example \(\PageIndex{55}\)
\((−4\sqrt{5})(5\sqrt{10})\)
Example \(\PageIndex{56}\)
\((5\sqrt{6})(−\sqrt{12})\)
- Answer
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\(−30\sqrt{2}\)
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 \(\PageIndex{60}\)
- \((\sqrt{15y})(\sqrt{5y^3})\)
- \((\sqrt{2n^2})(\sqrt{18n^3})\)
- Answer
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- \(5y^2\sqrt{3}\)
- \(6n^2\sqrt{n}\)
Example \(\PageIndex{61}\)
- \((\sqrt{14x^3})(\sqrt{7x^3})\)
- \((\sqrt{3q^2})(\sqrt{48q^3})\)
Example \(\PageIndex{62}\)
- \((\sqrt{16y^2})(\sqrt{8y^4})\)
- \((\sqrt{11s^6})(\sqrt{11s})\)
- Answer
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- \(8y^3\sqrt{2}\)
- \(11s^3\sqrt{s}\)
Example \(\PageIndex{63}\)
ⓐ \((\sqrt{8x^3})(\sqrt{3x})\)
ⓑ \((\sqrt{7r})(\sqrt{7r^8})\)
Example \(\PageIndex{64}\)
\((2\sqrt{5b^3})(4\sqrt{15b})\)
- Answer
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\(40b^2\sqrt{3}\)
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
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\(−96k^5\sqrt{k}\)
Example \(\PageIndex{71}\)
- \((\sqrt{7})^2\)
- \((−\sqrt{15})^2\)
Example \(\PageIndex{72}\)
- \((\sqrt{11})^2\)
- \((−\sqrt{21})^2\)
- Answer
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- 11
- 21
Example \(\PageIndex{73}\)
- \((\sqrt{19})^2\)
- \((−\sqrt{5})^2\)
Exercise \(\PageIndex{74}\)
- \((\sqrt{23})^2\)
- \((−\sqrt{3})^2\)
- Answer
-
- 23
- 3
Example \(\PageIndex{75}\)
- \((4\sqrt{11})(−3\sqrt{11})\)
- \((5\sqrt{3})^2\)
Example \(\PageIndex{76}\)
- \((2\sqrt{13})(−9\sqrt{13})\)
- \((6\sqrt{5})^2\)
- Answer
-
- −234
- 180
Example \(\PageIndex{77}\)
- \((−3\sqrt{12})(−2\sqrt{6})\)
- \( (−4\sqrt{10})^2\)
Example \(\PageIndex{78}\)
- \((−7\sqrt{5})(−3\sqrt{10})\)
- \( (−2\sqrt{14})^2\)
- Answer
-
- \(105\sqrt{2}\)
- 56
Use Polynomial Multiplication to Multiply Square Roots
In the following exercises, simplify.
Example \(\PageIndex{79}\)
- \(3(4−\sqrt{3})\)
- \(\sqrt{2}(4−\sqrt{6})\)
Example \(\PageIndex{80}\)
- \(4(6−\sqrt{11})\)
- \(\sqrt{2}(5−\sqrt{12})\)
- Answer
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- \(24−4\sqrt{11}\)
- \(5\sqrt{2}−2\sqrt{6}\)
Example \(\PageIndex{81}\)
- \(5(3−\sqrt{7})\)
- \(\sqrt{3}(4−\sqrt{15})\)
Example \(\PageIndex{82}\)
- \(7(−2−\sqrt{11})\)
- \(\sqrt{7}(6−\sqrt{14})\)
- Answer
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- \(−14−7\sqrt{11}\)
- \(6\sqrt{7}−7\sqrt{2}\)
Example \(\PageIndex{83}\)
- \(\sqrt{7}(5+2\sqrt{7})\)
- \(\sqrt{5}(\sqrt{10}+\sqrt{18})\)
Example \(\PageIndex{84}\)
- \(\sqrt{11}(8+4\sqrt{11})\)
- \(\sqrt{3}(\sqrt{12}+\sqrt{27})\)
- Answer
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- \(44+8\sqrt{11}\)
- 15
Example \(\PageIndex{85}\)
- \(\sqrt{11}(−3+4\sqrt{1})\)
- \(\sqrt{3}(\sqrt{15}−\sqrt{18})\)
Example \(\PageIndex{86}\)
- \(\sqrt{2}(−5+9\sqrt{2})\)
- \(\sqrt{7}(\sqrt{3}−\sqrt{21})\)
- Answer
-
- \(18−5\sqrt{2}\)
- \(\sqrt{21}−7\sqrt{3}\)
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
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\(52+3\sqrt{2}\)
Example \(\PageIndex{91}\)
\((3−\sqrt{7})(5−\sqrt{7})\)
Example \(\PageIndex{92}\)
\((5−\sqrt{7})(4−\sqrt{7})\)
- Answer
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\(27−9\sqrt{7}\)
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
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\(41+7\sqrt{55}\)
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 \(\PageIndex{103}\)
- \((3+\sqrt{5})^2\)
- \((2−5\sqrt{3})^2\)
Example \(\PageIndex{104}\)
- \((4+\sqrt{11})^2\)
- \((3−2\sqrt{5})^2\)
- Answer
-
- \(27+8\sqrt{11}\)
- \(29−12\sqrt{5}\)
Example \(\PageIndex{105}\)
- \((9−\sqrt{6})^2\)
- \((10+3\sqrt{7})^2\)
Example \(\PageIndex{106}\)
- \((5−\sqrt{10})^2\)
- \((8+3\sqrt{2})^2\)
- Answer
-
- \(35−10\sqrt{10}\)
- \(82+48\sqrt{2}\)
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.
- Use the Pythagorean Theorem to find the length of a side of the pool. Round your answer to the nearest tenth of a foot.
- Find the exact area of the pool.
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.
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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.
- Find the exact area of the face of the square base.
- Answer
-
- 4.6feet
- 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 \(\PageIndex{130}\)
- Explain why \((−\sqrt{n})^2\) is always positive, for \(n \ge 0\).
- Explain why \(−(\sqrt{n})^2\) is always negative, for \(n \ge 0\).
- Answer
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- when squaring a negative, it becomes a positive
- 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.
ⓑ 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?