13.8.3E: Exercises
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Practice Makes Perfect
Add and Subtract Like Square Roots
In the following exercises, simplify.
Example \(\PageIndex{46}\)
\(8\sqrt{2}−5\sqrt{2}\)
- Answer
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\(3\sqrt{2}\)
Example \(\PageIndex{47}\)
\(7\sqrt{2}−3\sqrt{2}\)
Example \(\PageIndex{48}\)
\(3\sqrt{5}+6\sqrt{5}\)
- Answer
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\(9\sqrt{5}\)
Example \(\PageIndex{49}\)
\(4\sqrt{5}+8\sqrt{5}\)
Example \(\PageIndex{50}\)
\(9\sqrt{7}−10\sqrt{7}\)
- Answer
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\(−\sqrt{7}\)
Example \(\PageIndex{51}\)
\(11\sqrt{7}−12\sqrt{7}\)
Example \(\PageIndex{52}\)
\(7\sqrt{y}+2\sqrt{y}\)
- Answer
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\(9\sqrt{y}\)
Example \(\PageIndex{53}\)
\(9\sqrt{n}+3\sqrt{n}\)
Example \(\PageIndex{54}\)
\(\sqrt{a}−4\sqrt{a}\)
- Answer
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\(−3\sqrt{a}\)
Example \(\PageIndex{55}\)
\(\sqrt{b}−6\sqrt{b}\)
Example \(\PageIndex{56}\)
\(5\sqrt{c}+2\sqrt{c}\)
- Answer
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\(7\sqrt{c}\)
Example \(\PageIndex{57}\)
\(7\sqrt{d}+2\sqrt{d}\)
Example \(\PageIndex{58}\)
\(8\sqrt{a}−2\sqrt{b}\)
- Answer
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\(8\sqrt{a}−2\sqrt{b}\)
Example \(\PageIndex{59}\)
\(5\sqrt{c}−3\sqrt{d}\)
Example \(\PageIndex{60}\)
\(5\sqrt{m}+\sqrt{n}\)
- Answer
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\(5\sqrt{m}+\sqrt{n}\)
Example \(\PageIndex{61}\)
\(\sqrt{n}+3\sqrt{p}\)
Example \(\PageIndex{62}\)
\(8\sqrt{7}+2\sqrt{7}+3\sqrt{7}\)
- Answer
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\(13\sqrt{7}\)
Example \(\PageIndex{63}\)
\(6\sqrt{5}+3\sqrt{5}+\sqrt{5}\)
Example \(\PageIndex{64}\)
\(3\sqrt{11}+2\sqrt{11}−8\sqrt{11}\)
- Answer
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\(−3\sqrt{11}\)
Example \(\PageIndex{65}\)
\(2\sqrt{15}+5\sqrt{15}−9\sqrt{15}\)
Example \(\PageIndex{66}\)
\(3\sqrt{3}−8\sqrt{3}+7\sqrt{5}\)
- Answer
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\(−5\sqrt{3}+7\sqrt{5}\)
Example \(\PageIndex{67}\)
\(5\sqrt{7}−8\sqrt{7}+6\sqrt{3}\)
Example \(\PageIndex{68}\)
\(6\sqrt{2}+2\sqrt{2}−3\sqrt{5}\)
- Answer
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\(8\sqrt{2}−3\sqrt{5}\)
Example \(\PageIndex{69}\)
\(7\sqrt{5}+\sqrt{5}−8\sqrt{10}\)
Example \(\PageIndex{70}\)
\(3\sqrt{2a}−4\sqrt{2a}+5\sqrt{2a}\)
- Answer
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\(4\sqrt{2a}\)
Example \(\PageIndex{71}\)
\(\sqrt{11b}−5\sqrt{11b}+3\sqrt{11b}\)
Example \(\PageIndex{72}\)
\(8\sqrt{3c}+2\sqrt{3c}−9\sqrt{3c}\)
- Answer
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\(\sqrt{3c}\)
Example \(\PageIndex{73}\)
\(3\sqrt{5d}+8\sqrt{5d}−11\sqrt{5d}\)
Example \(\PageIndex{74}\)
\(5\sqrt{3ab}+\sqrt{3ab}−2\sqrt{3ab}\)
- Answer
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\(4\sqrt{3ab}\
Example \(\PageIndex{75}\)
\(8\sqrt{11cd}+5\sqrt{11cd}−9\sqrt{11cd}\)
Example \(\PageIndex{76}\)
\(2\sqrt{pq}−5\sqrt{pq}+4\sqrt{pq}\)
- Answer
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\(\sqrt{pq}\)
Example \(\PageIndex{77}\)
\(11\sqrt{2rs}−9\sqrt{2rs}+3\sqrt{2rs}\)
In the following exercises, simplify.
Example \(\PageIndex{78}\)
\(\sqrt{50}+4\sqrt{2}\)
- Answer
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\(9\sqrt{2}\)
Example \(\PageIndex{79}\)
\(\sqrt{48}+2\sqrt{3}\)
Example \(\PageIndex{80}\)
\(\sqrt{80}−3\sqrt{5}\)
- Answer
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\(\sqrt{5}\)
Example \(\PageIndex{81}\)
\(\sqrt{28}−4\sqrt{7}\)
Example \(\PageIndex{82}\)
\(\sqrt{27}−\sqrt{75}\)
- Answer
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\(−2\sqrt{3}\)
Example \(\PageIndex{83}\)
\(\sqrt{72}−\sqrt{98}\)
Example \(\PageIndex{84}\)
\(\sqrt{48}+\sqrt{27}\)
- Answer
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\(7\sqrt{3}\)
Example \(\PageIndex{85}\)
\(\sqrt{45}+\sqrt{80}\)
Example \(\PageIndex{86}\)
\(2\sqrt{50}−3\sqrt{72}\)
- Answer
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\(−8\sqrt{2}\)
Example \(\PageIndex{87}\)
\(3\sqrt{98}−\sqrt{128}\)
Example \(\PageIndex{88}\)
\(2\sqrt{12}+3\sqrt{48}\)
- Answer
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\(16\sqrt{3}\)
Example \(\PageIndex{89}\)
\(4\sqrt{75}+2\sqrt{108}\)
Example \(\PageIndex{90}\)
\(\frac{2}{3}\sqrt{72}+\frac{1}{5}\sqrt{50}\)
- Answer
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\(5\sqrt{2}\)
Example \(\PageIndex{91}\)
\(\frac{2}{5}\sqrt{75}+\frac{3}{4}\sqrt{48}\)
Example \(\PageIndex{92}\)
\(\frac{1}{2}\sqrt{20}−\frac{2}{3}\sqrt{45}\)
- Answer
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\(−\sqrt{5}\)
Example \(\PageIndex{93}\)
\(\frac{2}{3}\sqrt{54}−\frac{3}{4}\sqrt{96}\)
Example \(\PageIndex{94}\)
\(\frac{1}{6}\sqrt{27}−\frac{3}{8}\sqrt{48}\)
- Answer
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\(−\sqrt{3}\)
Example \(\PageIndex{95}\)
\(\frac{1}{8}\sqrt{32}−\frac{1}{10}\sqrt{50}\)
Example \(\PageIndex{96}\)
\(\frac{1}{4}\sqrt{98}−\frac{1}{3}\sqrt{128}\)
- Answer
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\(−\frac{3}{4}\sqrt{2}\)
Example \(\PageIndex{97}\)
\(\frac{1}{3}\sqrt{24}+\frac{1}{4}\sqrt{54}\)
Example \(\PageIndex{98}\)
\(\sqrt{72a^5}−\sqrt{50a^5}\)
- Answer
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\(a^2\sqrt{2a}\)
Example \(\PageIndex{99}\)
\(\sqrt{48b^5}−\sqrt{75b^5}\)
Example \(\PageIndex{100}\)
\(\sqrt{80c^7}−\sqrt{20c^7}\)
- Answer
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\(2c^3\sqrt{5c}\)
Example \(\PageIndex{101}\)
\(\sqrt{96d^9}−\sqrt{24d^9}\)
Example \(\PageIndex{102}\)
\(9\sqrt{80p^4}−6\sqrt{98p^4}\)
- Answer
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\(36p^2\sqrt{5}−42p^2\sqrt{2}\)
Example \(\PageIndex{103}\)
\(8\sqrt{72q^6}−3\sqrt{75q^6}\)
Example \(\PageIndex{104}\)
\(2\sqrt{50r^8}+4\sqrt{54r^8}\)
- Answer
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\(10r^4\sqrt{2}+12r^4\sqrt{6}\)
Example \(\PageIndex{105}\)
\(5\sqrt{27s^6}+2\sqrt{20s^6}\)
Example \(\PageIndex{106}\)
\(3\sqrt{20x^2}−4\sqrt{45x^2}+5x\sqrt{80}\)
- Answer
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\(14x\sqrt{5}\)
Example \(\PageIndex{107}\)
\(2\sqrt{28x^2}−6\sqrt{3x^2}+6x\sqrt{7}\)
Example \(\PageIndex{108}\)
\(3\sqrt{128y^2}+4y\sqrt{162}−8\sqrt{98y^2}\)
- Answer
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\(−12y\sqrt{2}\)
Example \(\PageIndex{109}\)
\(3\sqrt{75y^2}+8y\sqrt{48}−\sqrt{300y^2}\)
Mixed Practice
Example \(\PageIndex{110}\)
\(2\sqrt{8}+6\sqrt{8}−5\sqrt{8}\)
- Answer
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\(3\sqrt{8}\)
Example \(\PageIndex{111}\)
\(\frac{2}{3}\sqrt{27}+\frac{3}{4}\sqrt{48}\)
Example \(\PageIndex{112}\)
\(\sqrt{175k^4}−\sqrt{63k^4}\)
- Answer
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\(2k^2\sqrt{7}\)
Example \(\PageIndex{113}\)
\(\frac{5}{6}\sqrt{162}+\frac{3}{16}\sqrt{128}\)
Example \(\PageIndex{114}\)
\(2\sqrt{363}−2\sqrt{300}\)
- Answer
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\(2\sqrt{3}\)
Example \(\PageIndex{115}\)
\(\sqrt{150}+4\sqrt{6}\)
Example \(\PageIndex{116}\)
\(9\sqrt{2}−8\sqrt{2}\)
- Answer
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\(\sqrt{2}\)
Example \(\PageIndex{117}\)
\(5\sqrt{x}−8\sqrt{y}\)
Example \(\PageIndex{118}\)
\(8\sqrt{13}−4\sqrt{13}−3\sqrt{13}\)
- Answer
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\(\sqrt{13}\)
Example \(\PageIndex{119}\)
\(5\sqrt{12c^4}−3\sqrt{27c^6}\)
Example \(\PageIndex{120}\)
\(\sqrt{80a^5}−\sqrt{45a^5}\)
- Answer
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\(a^2\sqrt{5a}\)
Example \(\PageIndex{121}\)
\(\frac{3}{5}\sqrt{75}−\frac{1}{4}\sqrt{48}\)
Example \(\PageIndex{122}\)
\(21\sqrt{19}−2\sqrt{19}\)
- Answer
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\(19\sqrt{19}\)
Example \(\PageIndex{123}\)
\(\sqrt{500}+\sqrt{405}\)
Example \(\PageIndex{124}\)
\(\frac{5}{6}\sqrt{27}+\frac{5}{8}\sqrt{48}\)
- Answer
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\(5\sqrt{3}\)
Example \(\PageIndex{125}\)
\(11\sqrt{11}−10\sqrt{11}\)
Example \(\PageIndex{126}\)
\(\sqrt{75}−\sqrt{108}\)
- Answer
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\(−\sqrt{3}\)
Example \(\PageIndex{127}\)
\(2\sqrt{98}−4\sqrt{72}\)
Example \(\PageIndex{128}\)
\(4\sqrt{24x^2}−\sqrt{54x^2}+3x\sqrt{6}\)
- Answer
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\(8x\sqrt{6}\)
Example \(\PageIndex{129}\)
\(8\sqrt{80y^6}−6\sqrt{48y^6}\)
Everyday Math
Example \(\PageIndex{130}\)
A decorator decides to use square tiles as an accent strip in the design of a new shower, but she wants to rotate the tiles to look like diamonds. She will use 9 large tiles that measure 8 inches on a side and 8 small tiles that measure 2 inches on a side. Determine the width of the accent strip by simplifying the expression \(9(8\sqrt{2})+8(2\sqrt{2})\). (Round to the nearest tenth of an inch.)
- Answer
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124.5 inches
Example \(\PageIndex{131}\)
Suzy wants to use square tiles on the border of a spa she is installing in her backyard. She will use large tiles that have area of 12 square inches, medium tiles that have area of 8 square inches, and small tiles that have area of 4 square inches. Once section of the border will require 4 large tiles, 8 medium tiles, and 10 small tiles to cover the width of the wall. Simplify the expression \(4\sqrt{12}+8\sqrt{8}+10\sqrt{4}\) to determine the width of the wall.
Writing Exercises
Example \(\PageIndex{132}\)
Explain the difference between like radicals and unlike radicals. Make sure your answer makes sense for radicals containing both numbers and variables.
- Answer
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Answers will vary.
Example \(\PageIndex{133}\)
Explain the process for determining whether two radicals are like or unlike. Make sure your answer makes sense for radicals containing both numbers and variables.
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?