Skip to main content
Mathematics LibreTexts

8.3E: Exercises

  • Page ID
    79518
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    Practice Makes Perfect

    Add and Subtract Like Square Roots

    In the following exercises, simplify.

    Example \(\PageIndex{46}\)

    \(8\sqrt{2}−5\sqrt{2}\)

    Answer

    \(3\sqrt{2}\)

    Example \(\PageIndex{47}\)

    \(7\sqrt{2}−3\sqrt{2}\)

    Example \(\PageIndex{48}\)

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

    Answer

    \(9\sqrt{5}\)

    Example \(\PageIndex{49}\)

    \(4\sqrt{5}+8\sqrt{5}\)

    Example \(\PageIndex{50}\)

    \(9\sqrt{7}−10\sqrt{7}\)

    Answer

    \(−\sqrt{7}\)

    Example \(\PageIndex{51}\)

    \(11\sqrt{7}−12\sqrt{7}\)

    Example \(\PageIndex{52}\)

    \(7\sqrt{y}+2\sqrt{y}\)

    Answer

    \(9\sqrt{y}\)

    Example \(\PageIndex{53}\)

    \(9\sqrt{n}+3\sqrt{n}\)

    Example \(\PageIndex{54}\)

    \(\sqrt{a}−4\sqrt{a}\)

    Answer

    \(−3\sqrt{a}\)

    Example \(\PageIndex{55}\)

    \(\sqrt{b}−6\sqrt{b}\)

    Example \(\PageIndex{56}\)

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

    Answer

    \(7\sqrt{c}\)

    Example \(\PageIndex{57}\)

    \(7\sqrt{d}+2\sqrt{d}\)

    Example \(\PageIndex{58}\)

    \(8\sqrt{a}−2\sqrt{b}\)

    Answer

    \(8\sqrt{a}−2\sqrt{b}\)

    Example \(\PageIndex{59}\)

    \(5\sqrt{c}−3\sqrt{d}\)

    Example \(\PageIndex{60}\)

    \(5\sqrt{m}+\sqrt{n}\)

    Answer

    \(5\sqrt{m}+\sqrt{n}\)

    Example \(\PageIndex{61}\)

    \(\sqrt{n}+3\sqrt{p}\)

    Example \(\PageIndex{62}\)

    \(8\sqrt{7}+2\sqrt{7}+3\sqrt{7}\)

    Answer

    \(13\sqrt{7}\)

    Example \(\PageIndex{63}\)

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

    Example \(\PageIndex{64}\)

    \(3\sqrt{11}+2\sqrt{11}−8\sqrt{11}\)

    Answer

    \(−3\sqrt{11}\)

    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

    \(−5\sqrt{3}+7\sqrt{5}\)

    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

    \(8\sqrt{2}−3\sqrt{5}\)

    Example \(\PageIndex{69}\)

    \(7\sqrt{5}+\sqrt{5}−8\sqrt{10}\)

    Example \(\PageIndex{70}\)

    \(3\sqrt{2a}−4\sqrt{2a}+5\sqrt{2a}\)

    Answer

    \(4\sqrt{2a}\)

    Example \(\PageIndex{71}\)

    \(\sqrt{11b}−5\sqrt{11b}+3\sqrt{11b}\)

    Example \(\PageIndex{72}\)

    \(8\sqrt{3c}+2\sqrt{3c}−9\sqrt{3c}\)

    Answer

    \(\sqrt{3c}\)

    Example \(\PageIndex{73}\)

    \(3\sqrt{5d}+8\sqrt{5d}−11\sqrt{5d}\)

    Example \(\PageIndex{74}\)

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

    Answer

    \(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

    \(\sqrt{pq}\)

    Example \(\PageIndex{77}\)

    \(11\sqrt{2rs}−9\sqrt{2rs}+3\sqrt{2rs}\)

    Add and Subtract Square Roots that Need Simplification

    In the following exercises, simplify.

    Example \(\PageIndex{78}\)

    \(\sqrt{50}+4\sqrt{2}\)

    Answer

    \(9\sqrt{2}\)

    Example \(\PageIndex{79}\)

    \(\sqrt{48}+2\sqrt{3}\)

    Example \(\PageIndex{80}\)

    \(\sqrt{80}−3\sqrt{5}\)

    Answer

    \(\sqrt{5}\)

    Example \(\PageIndex{81}\)

    \(\sqrt{28}−4\sqrt{7}\)

    Example \(\PageIndex{82}\)

    \(\sqrt{27}−\sqrt{75}\)

    Answer

    \(−2\sqrt{3}\)

    Example \(\PageIndex{83}\)

    \(\sqrt{72}−\sqrt{98}\)

    Example \(\PageIndex{84}\)

    \(\sqrt{48}+\sqrt{27}\)

    Answer

    \(7\sqrt{3}\)

    Example \(\PageIndex{85}\)

    \(\sqrt{45}+\sqrt{80}\)

    Example \(\PageIndex{86}\)

    \(2\sqrt{50}−3\sqrt{72}\)

    Answer

    \(−8\sqrt{2}\)

    Example \(\PageIndex{87}\)

    \(3\sqrt{98}−\sqrt{128}\)​​​​​​​

    Example \(\PageIndex{88}\)

    \(2\sqrt{12}+3\sqrt{48}\)

    Answer

    \(16\sqrt{3}\)

    Example \(\PageIndex{89}\)

    \(4\sqrt{75}+2\sqrt{108}\)

    Example \(\PageIndex{90}\)

    \(\frac{2}{3}\sqrt{72}+\frac{1}{5}\sqrt{50}\)

    Answer

    \(5\sqrt{2}\)

    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

    \(−\sqrt{5}\)

    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

    \(−\sqrt{3}\)​​​​​​​

    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

    \(−\frac{3}{4}\sqrt{2}\)

    Example \(\PageIndex{97}\)

    \(\frac{1}{3}\sqrt{24}+\frac{1}{4}\sqrt{54}\)

    Example \(\PageIndex{98}\)

    \(\sqrt{72a^5}−\sqrt{50a^5}\)

    Answer

    \(a^2\sqrt{2a}\)​​​​​​​

    Example \(\PageIndex{99}\)

    \(\sqrt{48b^5}−\sqrt{75b^5}\)​​​​​​​

    Example \(\PageIndex{100}\)

    \(\sqrt{80c^7}−\sqrt{20c^7}\)

    Answer

    \(2c^3\sqrt{5c}\)​​​​​​​

    Example \(\PageIndex{101}\)

    \(\sqrt{96d^9}−\sqrt{24d^9}\)

    Example \(\PageIndex{102}\)

    \(9\sqrt{80p^4}−6\sqrt{98p^4}\)

    Answer

    \(36p^2\sqrt{5}−42p^2\sqrt{2}\)​​​​​​​

    Example \(\PageIndex{103}\)

    \(8\sqrt{72q^6}−3\sqrt{75q^6}\)

    Example \(\PageIndex{104}\)

    \(2\sqrt{50r^8}+4\sqrt{54r^8}\)

    Answer

    \(10r^4\sqrt{2}+12r^4\sqrt{6}\)

    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

    \(14x\sqrt{5}\)

    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

    \(−12y\sqrt{2}\)

    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

    \(3\sqrt{8}\)​​​​​​​

    Example \(\PageIndex{111}\)

    \(\frac{2}{3}\sqrt{27}+\frac{3}{4}\sqrt{48}\)

    Example \(\PageIndex{112}\)

    \(\sqrt{175k^4}−\sqrt{63k^4}\)

    Answer

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

    Example \(\PageIndex{113}\)

    \(\frac{5}{6}\sqrt{162}+\frac{3}{16}\sqrt{128}\)

    Example \(\PageIndex{114}\)

    \(2\sqrt{363}−2\sqrt{300}\)

    Answer

    \(2\sqrt{3}\)​​​​​​​

    Example \(\PageIndex{115}\)

    \(\sqrt{150}+4\sqrt{6}\)

    Example \(\PageIndex{116}\)

    \(9\sqrt{2}−8\sqrt{2}\)

    Answer

    \(\sqrt{2}\)

    Example \(\PageIndex{117}\)

    \(5\sqrt{x}−8\sqrt{y}\)

    Example \(\PageIndex{118}\)

    \(8\sqrt{13}−4\sqrt{13}−3\sqrt{13}\)

    Answer

    \(\sqrt{13}\)​​​​​​​

    Example \(\PageIndex{119}\)

    \(5\sqrt{12c^4}−3\sqrt{27c^6}\)

    Example \(\PageIndex{120}\)

    \(\sqrt{80a^5}−\sqrt{45a^5}\)

    Answer

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

    Example \(\PageIndex{121}\)

    \(\frac{3}{5}\sqrt{75}−\frac{1}{4}\sqrt{48}\)

    Example \(\PageIndex{122}\)

    \(21\sqrt{19}−2\sqrt{19}\)

    Answer

    \(19\sqrt{19}\)

    Example \(\PageIndex{123}\)

    \(\sqrt{500}+\sqrt{405}\)

    Example \(\PageIndex{124}\)

    \(\frac{5}{6}\sqrt{27}+\frac{5}{8}\sqrt{48}\)

    Answer

    \(5\sqrt{3}\)

    Example \(\PageIndex{125}\)

    \(11\sqrt{11}−10\sqrt{11}\)​​​​​​​

    Example \(\PageIndex{126}\)

    \(\sqrt{75}−\sqrt{108}\)

    Answer

    \(−\sqrt{3}\)​​​​​​​

    Example \(\PageIndex{127}\)

    \(2\sqrt{98}−4\sqrt{72}\)

    Example \(\PageIndex{128}\)

    \(4\sqrt{24x^2}−\sqrt{54x^2}+3x\sqrt{6}\)

    Answer

    \(8x\sqrt{6}\)

    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

    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

    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.

    This table has four columns and three rows. The columns are labeled, “I can…,” “Confidently,” “With some help,” and “No – I don’t get it!” Under the “I can…” column the rows read, “add and subtract like square roots.,” and “add and subtract square roots that need simplification.” The other rows under the other columns are empty.

    ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?


    This page titled 8.3E: Exercises is shared under a CC BY license and was authored, remixed, and/or curated by OpenStax.

    • Was this article helpful?