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Mathematics LibreTexts

8.3E: Exercises

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    Practice Makes Perfect

    Exercise SET A: use the product property to simplify radical expressions

    In the following exercises, use the Product Property to simplify radical expressions.

    1. \(\sqrt{27}\)
    2. \(\sqrt{80}\)
    3. \(\sqrt{125}\)
    4. \(\sqrt{96}\)
    5. \(\sqrt{147}\)
    6. \(\sqrt{450}\)
    7. \(\sqrt{800}\)
    8. \(\sqrt{675}\)
      1. \(\sqrt[4]{32}\)
      2. \(\sqrt[5]{64}\)
      1. \(\sqrt[3]{625}\)
      2. \(\sqrt[6]{128}\)
      1. \(\sqrt[5]{64}\)
      2. \(\sqrt[3]{256}\)
      1. \(\sqrt[4]{3125}\)
      2. \(\sqrt[3]{81}\)
    Answer

    1. \(3\sqrt{3}\)

    3. \(5\sqrt{5}\)

    5. \(7\sqrt{3}\)

    7. \(20\sqrt{2}\)

    9.

    1. \(2 \sqrt[4]{2}\)
    2. \(2 \sqrt[5]{2}\)

    11.

    1. \(2 \sqrt[5]{2}\)
    2. \(4 \sqrt[3]{4}\)
    Exercise SET B: use the product property to simplify radical expressions

    In the following exercises, simplify using absolute value signs as needed.

      1. \(\sqrt{y^{11}}\)
      2. \(\sqrt[3]{r^{5}}\)
      3. \(\sqrt[4]{s^{10}}\)
      1. \(\sqrt{m^{13}}\)
      2. \(\sqrt[5]{u^{7}}\)
      3. \(\sqrt[6]{v^{11}}\)
      1. \(\sqrt{n^{21}}\)
      2. \(\sqrt[3]{q^{8}}\)
      3. \(\sqrt[8]{n^{10}}\)
      1. \(\sqrt{r^{25}}\)
      2. \(\sqrt[5]{p^{8}}\)
      3. \(\sqrt[4]{m^{5}}\)
      1. \(\sqrt{125 r^{13}}\)
      2. \(\sqrt[3]{108 x^{5}}\)
      3. \(\sqrt[4]{48 y^{6}}\)
      1. \(\sqrt{80 s^{15}}\)
      2. \(\sqrt[5]{96 a^{7}}\)
      3. \(\sqrt[6]{128 b^{7}}\)
      1. \(\sqrt{242 m^{23}}\)
      2. \(\sqrt[4]{405 m 10}\)
      3. \(\sqrt[5]{160 n^{8}}\)
      1. \(\sqrt{175 n^{13}}\)
      2. \(\sqrt[5]{512 p^{5}}\)
      3. \(\sqrt[4]{324 q^{7}}\)
      1. \(\sqrt{147 m^{7} n^{11}}\)
      2. \(\sqrt[3]{48 x^{6} y^{7}}\)
      3. \(\sqrt[4]{32 x^{5} y^{4}}\)
      1. \(\sqrt{96 r^{3} s^{3}}\)
      2. \(\sqrt[3]{80 x^{7} y^{6}}\)
      3. \(\sqrt[4]{80 x^{8} y^{9}}\)
      1. \(\sqrt{192 q^{3} r^{7}}\)
      2. \(\sqrt[3]{54 m^{9} n^{10}}\)
      3. \(\sqrt[4]{81 a^{9} b^{8}}\)
      1. \(\sqrt{150 m^{9} n^{3}}\)
      2. \(\sqrt[3]{81 p^{7} q^{8}}\)
      3. \(\sqrt[4]{162 c^{11} d^{12}}\)
      1. \(\sqrt[3]{-864}\)
      2. \(\sqrt[4]{-256}\)
      1. \(\sqrt[5]{-486}\)
      2. \(\sqrt[6]{-64}\)
      1. \(\sqrt[5]{-32}\)
      2. \(\sqrt[8]{-1}\)
      1. \(\sqrt[3]{-8}\)
      2. \(\sqrt[4]{-16}\)
      1. \(5+\sqrt{12}\)
      2. \(\dfrac{10-\sqrt{24}}{2}\)
      1. \(8+\sqrt{96}\)
      2. \(\dfrac{8-\sqrt{80}}{4}\)
      1. \(1+\sqrt{45}\)
      2. \(\dfrac{3+\sqrt{90}}{3}\)
      1. \(3+\sqrt{125}\)
      2. \(\dfrac{15+\sqrt{75}}{5}\)
    Answer

    1.

    1. \(\left|y^{5}\right| \sqrt{y}\)
    2. \(r \sqrt[3]{r^{2}}\)
    3. \(s^{2} \sqrt[4]{s^{2}}\)

    3.

    1. \(n^{10} \sqrt{n}\)
    2. \(q^{2} \sqrt[3]{q^{2}}\)
    3. \(|n| \sqrt[8]{n^{2}}\)

    5.

    1. \(5 r^{6} \sqrt{5 r}\)
    2. \(3 x \sqrt[3]{4 x^{2}}\)
    3. \(2|y| \sqrt[4]{3 y^{2}}\)

    7.

    1. \(11\left|m^{11}\right| \sqrt{2 m}\)
    2. \(3 m^{2} \sqrt[4]{5 m^{2}}\)
    3. \(2 n \sqrt[5]{5 n^{3}}\)

    9.

    1. \(7\left|m^{3} n^{5}\right| \sqrt{3 m n}\)
    2. \(2 x^{2} y^{2} \sqrt[3]{6 y}\)
    3. \(2|x y| \sqrt[4]{2 x}\)

    11.

    1. \(8\left|q r^{3}\right| \sqrt{3 q r}\)
    2. \(3 m^{3} n^{3} \sqrt[3]{2 n}\)
    3. \(3 a^{2} b^{2} \sqrt[4]{a}\)

    13.

    1. \(-6 \sqrt[3]{4}\)
    2. not real

    15.

    1. \(-2\)
    2. not real

    17.

    1. \(5+2 \sqrt{3}\)
    2. \(5-\sqrt{6}\)

    19.

    1. \(1+3 \sqrt{5}\)
    2. \(1+\sqrt{10}\)
    Exercise Set C: use the quotient property to simplify radical expressions

    In the following exercises, use the Quotient Property to simplify square roots.

      1. \(\sqrt{\dfrac{45}{80}}\)
      2. \(\sqrt[3]{\dfrac{8}{27}}\)
      3. \(\sqrt[4]{\dfrac{1}{81}}\)
      1. \(\sqrt{\dfrac{72}{98}}\)
      2. \(\sqrt[3]{\dfrac{24}{81}}\)
      3. \(\sqrt[4]{\dfrac{6}{96}}\)
      1. \(\sqrt{\dfrac{100}{36}}\)
      2. \(\sqrt[3]{\dfrac{81}{375}}\)
      3. \(\sqrt[4]{\dfrac{1}{256}}\)
      1. \(\sqrt{\dfrac{121}{16}}\)
      2. \(\sqrt[3]{\dfrac{16}{250}}\)
      3. \(\sqrt[4]{\dfrac{32}{162}}\)
      1. \(\sqrt{\dfrac{x^{10}}{x^{6}}}\)
      2. \(\sqrt[3]{\dfrac{p^{11}}{p^{2}}}\)
      3. \(\sqrt[4]{\dfrac{q^{17}}{q^{13}}}\)
      1. \(\sqrt{\dfrac{p^{20}}{p^{10}}}\)
      2. \(\sqrt[5]{\dfrac{d^{12}}{d^{7}}}\)
      3. \(\sqrt[8]{\dfrac{m^{12}}{m^{4}}}\)
      1. \(\sqrt{\dfrac{y^{4}}{y^{8}}}\)
      2. \(\sqrt[5]{\dfrac{u^{21}}{u^{11}}}\)
      3. \(\sqrt[6]{\dfrac{v^{30}}{v^{12}}}\)
      1. \(\sqrt{\dfrac{q^{8}}{q^{14}}}\)
      2. \(\sqrt[3]{\dfrac{r^{14}}{r^{5}}}\)
      3. \(\sqrt[4]{\dfrac{c^{21}}{c^{9}}}\)
    1. \(\sqrt{\dfrac{96 x^{7}}{121}}\)
    2. \(\sqrt{\dfrac{108 y^{4}}{49}}\)
    3. \(\sqrt{\dfrac{300 m^{5}}{64}}\)
    4. \(\sqrt{\dfrac{125 n^{7}}{169}}\)
    5. \(\sqrt{\dfrac{98 r^{5}}{100}}\)
    6. \(\sqrt{\dfrac{180 s^{10}}{144}}\)
    7. \(\sqrt{\dfrac{28 q^{6}}{225}}\)
    8. \(\sqrt{\dfrac{150 r^{3}}{256}}\)
      1. \(\sqrt{\dfrac{75 r^{9}}{s^{8}}}\)
      2. \(\sqrt[3]{\dfrac{54 a^{8}}{b^{3}}}\)
      3. \(\sqrt[4]{\dfrac{64 c^{5}}{d^{4}}}\)
      1. \(\sqrt{\dfrac{72 x^{5}}{y^{6}}}\)
      2. \(\sqrt[5]{\dfrac{96 r^{11}}{s^{5}}}\)
      3. \(\sqrt[6]{\dfrac{128 u^{7}}{v^{12}}}\)
      1. \(\sqrt{\dfrac{28 p^{7}}{q^{2}}}\)
      2. \(\sqrt[3]{\dfrac{81 s^{8}}{t^{3}}}\)
      3. \(\sqrt[4]{\dfrac{64 p^{15}}{q^{12}}}\)
      1. \(\sqrt{\dfrac{45 r^{3}}{s^{10}}}\)
      2. \(\sqrt[3]{\dfrac{625 u^{10}}{v^{3}}}\)
      3. \(\sqrt[4]{\dfrac{729 c^{21}}{d^{8}}}\)
      1. \(\sqrt{\dfrac{32 x^{5} y^{3}}{18 x^{3} y}}\)
      2. \(\sqrt[3]{\dfrac{5 x^{6} y^{9}}{40 x^{5} y^{3}}}\)
      3. \(\sqrt[4]{\dfrac{5 a^{8} b^{6}}{80 a^{3} b^{2}}}\)
      1. \(\sqrt{\dfrac{75 r^{6} s^{8}}{48 r s^{4}}}\)
      2. \(\sqrt[3]{\dfrac{24 x^{8} y^{4}}{81 x^{2} y}}\)
      3. \(\sqrt[4]{\dfrac{32 m^{9} n^{2}}{162 m n^{2}}}\)
      1. \(\sqrt{\dfrac{27 p^{2} q}{108 p^{4} q^{3}}}\)
      2. \(\sqrt[3]{\dfrac{16 c^{5} d^{7}}{250 c^{2} d^{2}}}\)
      3. \(\sqrt[6]{\dfrac{2 m^{9} n^{7}}{128 m^{3} n}}\)
      1. \(\sqrt{\dfrac{50 r^{5} s^{2}}{128 r^{2} s^{6}}}\)
      2. \(\sqrt[3]{\dfrac{24 m^{9} n^{7}}{375 m^{4} n}}\)
      3. \(\sqrt[4]{\dfrac{81 m^{2} n^{8}}{256 m^{1} n^{2}}}\)
      1. \(\dfrac{\sqrt{45 p^{9}}}{\sqrt{5 q^{2}}}\)
      2. \(\dfrac{\sqrt[4]{64}}{\sqrt[4]{2}}\)
      3. \(\dfrac{\sqrt[5]{128 x^{8}}}{\sqrt[5]{2 x^{2}}}\)
      1. \(\dfrac{\sqrt{80 q^{5}}}{\sqrt{5 q}}\)
      2. \(\dfrac{\sqrt[3]{-625}}{\sqrt[3]{5}}\)
      3. \(\dfrac{\sqrt[4]{80 m^{7}}}{\sqrt[4]{5 m}}\)
      1. \(\dfrac{\sqrt{50 m^{7}}}{\sqrt{2 m}}\)
      2. \(\sqrt[3]{\dfrac{1250}{2}}\)
      3. \(\sqrt[4]{\dfrac{486 y^{9}}{2 y^{3}}}\)
      1. \(\dfrac{\sqrt{72 n^{11}}}{\sqrt{2 n}}\)
      2. \(\sqrt[3]{\dfrac{162}{6}}\)
      3. \(\sqrt[4]{\dfrac{160 r^{10}}{5 r^{3}}}\)
    Answer

    1.

    1. \(\dfrac{3}{4}\)
    2. \(\dfrac{2}{3}\)
    3. \(\dfrac{1}{3}\)

    3.

    1. \(\dfrac{5}{3}\)
    2. \(\dfrac{3}{5}\)
    3. \(\dfrac{1}{4}\)

    5.

    1. \(x^{2}\)
    2. \(p^{3}\)
    3. \(|q|\)

    7.

    1. \(\dfrac{1}{y^{2}}\)
    2. \(u^{2}\)
    3. \(|v^{3}|\)

    9. \(\dfrac{4\left|x^{3}\right| \sqrt{6 x}}{11}\)

    11. \(\dfrac{10 m^{2} \sqrt{3 m}}{8}\)

    13. \(\dfrac{7 r^{2} \sqrt{2 r}}{10}\)

    15. \(\dfrac{2\left|q^{3}\right| \sqrt{7}}{15}\)

    17.

    1. \(\dfrac{5 r^{4} \sqrt{3 r}}{s^{4}}\)
    2. \(\dfrac{3 a^{2} \sqrt[3]{2 a^{2}}}{|b|}\)
    3. \(\dfrac{2|c| \sqrt[4]{4 c}}{|d|}\)

    19.

    1. \(\dfrac{2\left|p^{3}\right| \sqrt{7 p}}{|q|}\)
    2. \(\dfrac{3 s^{2} \sqrt[3]{3 s^{2}}}{t}\)
    3. \(\dfrac{2\left|p^{3}\right| \sqrt[4]{4 p^{3}}}{\left|q^{3}\right|}\)

    21.

    1. \(\dfrac{4|x y|}{3}\)
    2. \(\dfrac{y^{2} \sqrt[3]{x}}{2}\)
    3. \(\dfrac{|a b| \sqrt[4]{a}}{4}\)

    23.

    1. \(\dfrac{1}{2|p q|}\)
    2. \(\dfrac{2 c d \sqrt[5]{2 d^{2}}}{5}\)
    3. \(\dfrac{|m n| \sqrt[6]{2}}{2}\)

    25.

    1. \(\dfrac{3 p^{4} \sqrt{p}}{|q|}\)
    2. \(2 \sqrt[4]{2}\)
    3. \(2 x \sqrt[5]{2 x}\)

    27.

    1. \(5\left|m^{3}\right|\)
    2. \(5 \sqrt[3]{5}\)
    3. \(3|y| \sqrt[4]{3 y^{2}}\)
    Exercise SET D: writing exercises
    1. Explain why \(\sqrt{x^{4}}=x^{2}\). Then explain why \(\sqrt{x^{16}}=x^{8}\).
    2. Explain why \(7+\sqrt{9}\) is not equal to \(\sqrt{7+9}\).
    3. Explain how you know that \(\sqrt[5]{x^{10}}=x^{2}\).
    4. Explain why \(\sqrt[4]{-64}\) is not a real number but \(\sqrt[3]{-64}\) is.
    Answer

    1. Answers may vary

    3. Answers may vary

    Self Check

    a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

    This table has 3 rows and 4 columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is “Confidently”, the third is “With some help”, and the fourth is “No, I don’t get it”. Under the first column are the phrases “use the product property to simplify radical expressions” and “use the quotient property to simplify radical expressions”. The other columns are left blank so that the learner may indicate their mastery level for each topic.
    Figure 8.2.1

    b. After reviewing this checklist, what will you do to become confident for all objectives?

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