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9.5E: Exercises

  • Page ID
    30279
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    Practice Makes Perfect

    Divide Square Roots

    In the following exercises, simplify.

    Example \(\PageIndex{43}\)

    \(\frac{\sqrt{27}}{6}\)

    Answer

    \(\frac{\sqrt{3}}{2}\)

    Example \(\PageIndex{44}\)

    \(\frac{\sqrt{50}}{10}\)

    Example \(\PageIndex{45}\)

    \(\frac{\sqrt{72}}{9}\)

    Answer

    \(\frac{2\sqrt{2}}{3}\)

    Example \(\PageIndex{46}\)

    \(\frac{\sqrt{243}}{6}\)

    Example \(\PageIndex{47}\)

    \(\frac{2−\sqrt{32}}{8}\)

    Answer

    \(\frac{1−2\sqrt{2}}{4}\)

    Example \(\PageIndex{48}\)

    \(\frac{3+\sqrt{27}}{9}\)

    Example \(\PageIndex{49}\)

    \(\frac{6+\sqrt{45}}{6}\)

    Answer

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

    Example \(\PageIndex{50}\)

    \(\frac{10−\sqrt{200}}{20}\)

    Example \(\PageIndex{51}\)

    \(\frac{\sqrt{80}}{\sqrt{125}}\)

    Answer

    \(\frac{4}{5}\)

    Example \(\PageIndex{52}\)

    \(\frac{\sqrt{72}}{\sqrt{200}}\)

    Example \(\PageIndex{53}\)

    \(\frac{\sqrt{128}}{\sqrt{72}}\)

    Answer

    \(\frac{4}{3}\)

    Example \(\PageIndex{54}\)

    \(\frac{\sqrt{48}}{\sqrt{75}}\)

    Example \(\PageIndex{55}\)
    1. \(\frac{\sqrt{8x^6}}{2x^2}\)
    2. \(\frac{\sqrt{200m^5}}{98m}\)
    Answer
    1. \(2x^2\)
    2. \(\frac{10m^2}{7}\)
    Example \(\PageIndex{56}\)
    1. \(\frac{\sqrt{10y^3}}{5y}\)
    2. \(\frac{\sqrt{108n^7}}{243n^3}\)
    Example \(\PageIndex{57}\)

    \(\frac{\sqrt{75r^3}}{108r}\)

    Answer

    \(\frac{5r}{6}\)

    Example \(\PageIndex{58}\)

    \(\frac{\sqrt{196q^5}}{484q}\)

    Example \(\PageIndex{59}\)

    \(\frac{\sqrt{108p^{5}q^{2}}}{\sqrt{34p^{3}q^{6}}}\)

    Answer

    \(\frac{3p\sqrt{102}}{17q^2}\)

    Example \(\PageIndex{60}\)

    \(\frac{\sqrt{98rs^{10}}}{\sqrt{2r^{3}s^{4}}}\)

    Example \(\PageIndex{61}\)

    \(\frac{\sqrt{320mn^{5}}}{\sqrt{45m^{7}n^{3}}}\)

    Answer

    \(\frac{8n}{3m^3}\)

    Example \(\PageIndex{62}\)

    \(\frac{\sqrt{810c^{3}d^{7}}}{\sqrt{1000c^{5}d}}\)

    Example \(\PageIndex{63}\)

    \(\frac{\sqrt{98}}{14}\)

    Answer

    \(\frac{\sqrt{2}}{2}\)

    Example \(\PageIndex{64}\)

    \(\frac{\sqrt{72}}{18}\)

    Example \(\PageIndex{65}\)

    \(\frac{5+\sqrt{125}}{15}\)

    Answer

    \(\frac{1+\sqrt{3}}{3}\)

    Example \(\PageIndex{66}\)

    \(\frac{6−\sqrt{45}}{12}\)

    Example \(\PageIndex{67}\)

    \(\frac{\sqrt{96}}{\sqrt{150}}\)

    Answer

    \(\frac{4}{5}\)

    Example \(\PageIndex{68}\)

    \(\frac{\sqrt{28}}{\sqrt{63}}\)

    Example \(\PageIndex{69}\)

    \(\frac{\sqrt{26y^7}}{2y}\)

    Answer

    \(y^3\sqrt{13}\)

    Example \(\PageIndex{70}\)

    \(\frac{\sqrt{15x^3}}{\sqrt{3x}}\)

    Rationalize a One-Term Denominator

    In the following exercises, simplify and rationalize the denominator.

    Example \(\PageIndex{71}\)

    \(\frac{10}{\sqrt{6}}\)

    Answer

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

    Example \(\PageIndex{72}\)

    \(\frac{8}{\sqrt{3}}\)

    Example \(\PageIndex{73}\)

    \(\frac{6}{\sqrt{7}}\)

    Answer

    \(\frac{6\sqrt{7}}{7}\)

    Example \(\PageIndex{74}\)

    \(\frac{4}{\sqrt{5}}\)

    Example \(\PageIndex{75}\)

    \(\frac{3}{\sqrt{13}}\)

    Answer

    \(\frac{3\sqrt{13}}{13}\)

    Example \(\PageIndex{76}\)

    \(\frac{10}{\sqrt{11}}\)

    Example \(\PageIndex{77}\)

    \(\frac{10}{3\sqrt{10}}\)

    Answer

    \(\frac{\sqrt{10}}{3}\)

    Example \(\PageIndex{78}\)

    \(\frac{2}{5\sqrt{2}}\)

    Example \(\PageIndex{79}\)

    \(\frac{4}{9\sqrt{5}}\)

    Answer

    \(\frac{4\sqrt{5}}{45}\)

    Example \(\PageIndex{80}\)

    \(\frac{9}{2\sqrt{7}}\)

    Example \(\PageIndex{81}\)

    \(−\frac{9}{2\sqrt{3}}\)

    Answer

    \(−\frac{3\sqrt{3}}{2}\)

    Example \(\PageIndex{82}\)

    \(−\frac{8}{3\sqrt{6}}\)

    Example \(\PageIndex{83}\)

    \(\sqrt{\frac{3}{20}}\)

    Answer

    \(\frac{\sqrt{15}}{10}\)

    Example \(\PageIndex{84}\)

    \(\sqrt{\frac{4}{27}}\)

    Example \(\PageIndex{85}\)

    \(\sqrt{\frac{7}{40}}\)

    Answer

    \(\frac{\sqrt{70}}{20}\)

    Example \(\PageIndex{86}\)

    \(\sqrt{\frac{8}{45}}\)

    Example \(\PageIndex{87}\)

    \(\sqrt{\frac{19}{175}}\)

    Answer

    \(\frac{\sqrt{133}}{35}\)

    Example \(\PageIndex{88}\)

    \(\sqrt{\frac{17}{192}}\)

    Rationalize a Two-Term Denominator

    In the following exercises, simplify by rationalizing the denominator.

    Example \(\PageIndex{89}\)
    1. \(\frac{3}{3+\sqrt{11}}\)
    2. \(\frac{8}{1−\sqrt{5}}\)
    Answer
    1. \(\frac{3(3−\sqrt{11})}{−2}\)
    2. \(−2(1+\sqrt{5})\)
    Example \(\PageIndex{90}\)
    1. \(\frac{4}{4+\sqrt{7}}\)
    2. \(\frac{7}{2−\sqrt{6}}\)
    Example \(\PageIndex{91}\)
    1. \(\frac{5}{5+\sqrt{6}}\)
    2. \(\frac{6}{3−\sqrt{7}}\)
    Answer
    1. \(\frac{5(5−\sqrt{6})}{19}\)
    2. \(3(3+\sqrt{7})\)
    Example \(\PageIndex{92}\)
    1. \(\frac{6}{6+\sqrt{5}}\)
    2. \(\frac{5}{4−\sqrt{11}}\)
    Example \(\PageIndex{93}\)

    \(\frac{\sqrt{3}}{\sqrt{m}−\sqrt{5}}\)

    Answer

    \(\frac{\sqrt{3}(\sqrt{m}+\sqrt{5})}{m−5}\)

    Example \(\PageIndex{94}\)

    \(\frac{\sqrt{5}}{\sqrt{n}−\sqrt{7}}\)

    Example \(\PageIndex{95}\)

    \(\frac{\sqrt{2}}{\sqrt{x}−\sqrt{6}}\)

    Answer

    \(\frac{\sqrt{2}(\sqrt{x}+\sqrt{3})}{x−6}\)

    Example \(\PageIndex{96}\)

    \(\frac{\sqrt{7}}{\sqrt{y}+\sqrt{3}}\)

    Example \(\PageIndex{97}\)

    \(\frac{\sqrt{r}+\sqrt{5}}{\sqrt{r}−\sqrt{5}}\)

    Answer

    \(\frac{(\sqrt{r}+\sqrt{5})^2}{r−5}\)

    Example \(\PageIndex{98}\)

    \(\frac{\sqrt{s}−\sqrt{6}}{\sqrt{s}+\sqrt{6}}\)

    Example \(\PageIndex{99}\)

    \(\frac{\sqrt{150x^{2}y^{6}}}{\sqrt{6x^{4}y^{2}}}\)

    Answer

    \(\frac{5y^2}{x}\)

    Example \(\PageIndex{100}\)

    \(\frac{\sqrt{80p^{3}q}}{\sqrt{5pq^{5}}}\)

    Example \(\PageIndex{101}\)

    \(\frac{15}{\sqrt{5}}\)

    Answer

    \(3\sqrt{5}\)

    Example \(\PageIndex{102}\)

    \(\frac{3}{5\sqrt{8}}\)

    Example \(\PageIndex{103}\)

    \(\sqrt{\frac{8}{54}}\)

    Answer

    \(\frac{2\sqrt{3}}{9}\)

    Example \(\PageIndex{104}\)

    \(\sqrt{\frac{12}{20}}\)

    Example \(\PageIndex{105}\)

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

    Answer

    \(\frac{3(5−\sqrt{5})}{20}\)

    Example \(\PageIndex{106}\)

    \(\frac{20}{4−\sqrt{3}}\)

    Example \(\PageIndex{107}\)

    \(\frac{\sqrt{2}}{\sqrt{x}−\sqrt{3}}\)

    Answer

    \(\frac{\sqrt{2}(\sqrt{x}+\sqrt{3})}{x−3}\)

    Example \(\PageIndex{108}\)

    \(\frac{\sqrt{5}}{\sqrt{y}−\sqrt{7}}\)

    Example \(\PageIndex{109}\)

    \(\frac{\sqrt{x}+\sqrt{8}}{\sqrt{x}−\sqrt{8}}\)

    Answer

    \(\frac{(\sqrt{x}+2\sqrt{2})^2}{x−8}\)

    Example \(\PageIndex{110}\)

    \(\frac{\sqrt{m}−\sqrt{3}}{\sqrt{m}+\sqrt{3}}\)

    Everyday Math

    Example \(\PageIndex{111}\)

    A supply kit is dropped from an airplane flying at an altitude of 250 feet. Simplify \(\sqrt{\frac{250}{16}}\) to determine how many seconds it takes for the supply kit to reach the ground.

    Answer

    \(\frac{5\sqrt{10}}{4}\) seconds

    Example \(\PageIndex{112}\)

    A flare is dropped into the ocean from an airplane flying at an altitude of 1,200 feet. Simplify \(\sqrt{\frac{1200}{16}}\) to determine how many seconds it takes for the flare to reach the ocean.

    Writing Exercises

    Example \(\PageIndex{113}\)
    1. Simplify \(\sqrt{\frac{27}{3}}\) and explain all your steps.
    2. Simplify \(\sqrt{\frac{27}{5}}\) and explain all your steps.
    3. Why are the two methods of simplifying square roots different?
    Answer

    Answers will vary.

    Example \(\PageIndex{114}\)
    1. Approximate \(\frac{1}{\sqrt{2}}\) by dividing \(\frac{1}{1.414}\) using long division without a calculator.
    2. Rationalizing the denominator of \(\frac{1}{\sqrt{2}}\) gives \(\frac{\sqrt{2}}{2}\). Approximate \(\frac{\sqrt{2}}{2}\) by dividing \(\frac{1.414}{2}\) using long division without a calculator.
    3. Do you agree that rationalizing the denominator makes calculations easier? Why or why not?

    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 four rows. The columns are labeled, “I can…,” “confidently.,” “with some help.,” and “no – I don’t get it!” The rows under the column “I can…” read, “divide square roots,” “rationalize a one term denominator.,” and “rationalize a two term denominator.” All the other rows under the columns are empty.

    ⓑ After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?


    This page titled 9.5E: Exercises is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.