8.3E: Exercises
Practice Makes Perfect
In the following exercises, use the Product Property to simplify radical expressions.
- \(\sqrt{27}\)
- \(\sqrt{80}\)
- \(\sqrt{125}\)
- \(\sqrt{96}\)
- \(\sqrt{147}\)
- \(\sqrt{450}\)
- \(\sqrt{800}\)
- \(\sqrt{675}\)
-
- \(\sqrt[4]{32}\)
- \(\sqrt[5]{64}\)
-
- \(\sqrt[3]{625}\)
- \(\sqrt[6]{128}\)
-
- \(\sqrt[5]{64}\)
- \(\sqrt[3]{256}\)
-
- \(\sqrt[4]{3125}\)
- \(\sqrt[3]{81}\)
- Answer
-
1. \(3\sqrt{3}\)
3. \(5\sqrt{5}\)
5. \(7\sqrt{3}\)
7. \(20\sqrt{2}\)
9.
- \(2 \sqrt[4]{2}\)
- \(2 \sqrt[5]{2}\)
11.
- \(2 \sqrt[5]{2}\)
- \(4 \sqrt[3]{4}\)
In the following exercises, simplify using absolute value signs as needed.
-
- \(\sqrt{y^{11}}\)
- \(\sqrt[3]{r^{5}}\)
- \(\sqrt[4]{s^{10}}\)
-
- \(\sqrt{m^{13}}\)
- \(\sqrt[5]{u^{7}}\)
- \(\sqrt[6]{v^{11}}\)
-
- \(\sqrt{n^{21}}\)
- \(\sqrt[3]{q^{8}}\)
- \(\sqrt[8]{n^{10}}\)
-
- \(\sqrt{r^{25}}\)
- \(\sqrt[5]{p^{8}}\)
- \(\sqrt[4]{m^{5}}\)
-
- \(\sqrt{125 r^{13}}\)
- \(\sqrt[3]{108 x^{5}}\)
- \(\sqrt[4]{48 y^{6}}\)
-
- \(\sqrt{80 s^{15}}\)
- \(\sqrt[5]{96 a^{7}}\)
- \(\sqrt[6]{128 b^{7}}\)
-
- \(\sqrt{242 m^{23}}\)
- \(\sqrt[4]{405 m 10}\)
- \(\sqrt[5]{160 n^{8}}\)
-
- \(\sqrt{175 n^{13}}\)
- \(\sqrt[5]{512 p^{5}}\)
- \(\sqrt[4]{324 q^{7}}\)
-
- \(\sqrt{147 m^{7} n^{11}}\)
- \(\sqrt[3]{48 x^{6} y^{7}}\)
- \(\sqrt[4]{32 x^{5} y^{4}}\)
-
- \(\sqrt{96 r^{3} s^{3}}\)
- \(\sqrt[3]{80 x^{7} y^{6}}\)
- \(\sqrt[4]{80 x^{8} y^{9}}\)
-
- \(\sqrt{192 q^{3} r^{7}}\)
- \(\sqrt[3]{54 m^{9} n^{10}}\)
- \(\sqrt[4]{81 a^{9} b^{8}}\)
-
- \(\sqrt{150 m^{9} n^{3}}\)
- \(\sqrt[3]{81 p^{7} q^{8}}\)
- \(\sqrt[4]{162 c^{11} d^{12}}\)
-
- \(\sqrt[3]{-864}\)
- \(\sqrt[4]{-256}\)
-
- \(\sqrt[5]{-486}\)
- \(\sqrt[6]{-64}\)
-
- \(\sqrt[5]{-32}\)
- \(\sqrt[8]{-1}\)
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- \(\sqrt[3]{-8}\)
- \(\sqrt[4]{-16}\)
-
- \(5+\sqrt{12}\)
- \(\dfrac{10-\sqrt{24}}{2}\)
-
- \(8+\sqrt{96}\)
- \(\dfrac{8-\sqrt{80}}{4}\)
-
- \(1+\sqrt{45}\)
- \(\dfrac{3+\sqrt{90}}{3}\)
-
- \(3+\sqrt{125}\)
- \(\dfrac{15+\sqrt{75}}{5}\)
- Answer
-
1.
- \(\left|y^{5}\right| \sqrt{y}\)
- \(r \sqrt[3]{r^{2}}\)
- \(s^{2} \sqrt[4]{s^{2}}\)
3.
- \(n^{10} \sqrt{n}\)
- \(q^{2} \sqrt[3]{q^{2}}\)
- \(|n| \sqrt[8]{n^{2}}\)
5.
- \(5 r^{6} \sqrt{5 r}\)
- \(3 x \sqrt[3]{4 x^{2}}\)
- \(2|y| \sqrt[4]{3 y^{2}}\)
7.
- \(11\left|m^{11}\right| \sqrt{2 m}\)
- \(3 m^{2} \sqrt[4]{5 m^{2}}\)
- \(2 n \sqrt[5]{5 n^{3}}\)
9.
- \(7\left|m^{3} n^{5}\right| \sqrt{3 m n}\)
- \(2 x^{2} y^{2} \sqrt[3]{6 y}\)
- \(2|x y| \sqrt[4]{2 x}\)
11.
- \(8\left|q r^{3}\right| \sqrt{3 q r}\)
- \(3 m^{3} n^{3} \sqrt[3]{2 n}\)
- \(3 a^{2} b^{2} \sqrt[4]{a}\)
13.
- \(-6 \sqrt[3]{4}\)
- not real
15.
- \(-2\)
- not real
17.
- \(5+2 \sqrt{3}\)
- \(5-\sqrt{6}\)
19.
- \(1+3 \sqrt{5}\)
- \(1+\sqrt{10}\)
In the following exercises, use the Quotient Property to simplify square roots.
-
- \(\sqrt{\dfrac{45}{80}}\)
- \(\sqrt[3]{\dfrac{8}{27}}\)
- \(\sqrt[4]{\dfrac{1}{81}}\)
-
- \(\sqrt{\dfrac{72}{98}}\)
- \(\sqrt[3]{\dfrac{24}{81}}\)
- \(\sqrt[4]{\dfrac{6}{96}}\)
-
- \(\sqrt{\dfrac{100}{36}}\)
- \(\sqrt[3]{\dfrac{81}{375}}\)
- \(\sqrt[4]{\dfrac{1}{256}}\)
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- \(\sqrt{\dfrac{121}{16}}\)
- \(\sqrt[3]{\dfrac{16}{250}}\)
- \(\sqrt[4]{\dfrac{32}{162}}\)
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- \(\sqrt{\dfrac{x^{10}}{x^{6}}}\)
- \(\sqrt[3]{\dfrac{p^{11}}{p^{2}}}\)
- \(\sqrt[4]{\dfrac{q^{17}}{q^{13}}}\)
-
- \(\sqrt{\dfrac{p^{20}}{p^{10}}}\)
- \(\sqrt[5]{\dfrac{d^{12}}{d^{7}}}\)
- \(\sqrt[8]{\dfrac{m^{12}}{m^{4}}}\)
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- \(\sqrt{\dfrac{y^{4}}{y^{8}}}\)
- \(\sqrt[5]{\dfrac{u^{21}}{u^{11}}}\)
- \(\sqrt[6]{\dfrac{v^{30}}{v^{12}}}\)
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- \(\sqrt{\dfrac{q^{8}}{q^{14}}}\)
- \(\sqrt[3]{\dfrac{r^{14}}{r^{5}}}\)
- \(\sqrt[4]{\dfrac{c^{21}}{c^{9}}}\)
- \(\sqrt{\dfrac{96 x^{7}}{121}}\)
- \(\sqrt{\dfrac{108 y^{4}}{49}}\)
- \(\sqrt{\dfrac{300 m^{5}}{64}}\)
- \(\sqrt{\dfrac{125 n^{7}}{169}}\)
- \(\sqrt{\dfrac{98 r^{5}}{100}}\)
- \(\sqrt{\dfrac{180 s^{10}}{144}}\)
- \(\sqrt{\dfrac{28 q^{6}}{225}}\)
- \(\sqrt{\dfrac{150 r^{3}}{256}}\)
-
- \(\sqrt{\dfrac{75 r^{9}}{s^{8}}}\)
- \(\sqrt[3]{\dfrac{54 a^{8}}{b^{3}}}\)
- \(\sqrt[4]{\dfrac{64 c^{5}}{d^{4}}}\)
-
- \(\sqrt{\dfrac{72 x^{5}}{y^{6}}}\)
- \(\sqrt[5]{\dfrac{96 r^{11}}{s^{5}}}\)
- \(\sqrt[6]{\dfrac{128 u^{7}}{v^{12}}}\)
-
- \(\sqrt{\dfrac{28 p^{7}}{q^{2}}}\)
- \(\sqrt[3]{\dfrac{81 s^{8}}{t^{3}}}\)
- \(\sqrt[4]{\dfrac{64 p^{15}}{q^{12}}}\)
-
- \(\sqrt{\dfrac{45 r^{3}}{s^{10}}}\)
- \(\sqrt[3]{\dfrac{625 u^{10}}{v^{3}}}\)
- \(\sqrt[4]{\dfrac{729 c^{21}}{d^{8}}}\)
-
- \(\sqrt{\dfrac{32 x^{5} y^{3}}{18 x^{3} y}}\)
- \(\sqrt[3]{\dfrac{5 x^{6} y^{9}}{40 x^{5} y^{3}}}\)
- \(\sqrt[4]{\dfrac{5 a^{8} b^{6}}{80 a^{3} b^{2}}}\)
-
- \(\sqrt{\dfrac{75 r^{6} s^{8}}{48 r s^{4}}}\)
- \(\sqrt[3]{\dfrac{24 x^{8} y^{4}}{81 x^{2} y}}\)
- \(\sqrt[4]{\dfrac{32 m^{9} n^{2}}{162 m n^{2}}}\)
-
- \(\sqrt{\dfrac{27 p^{2} q}{108 p^{4} q^{3}}}\)
- \(\sqrt[3]{\dfrac{16 c^{5} d^{7}}{250 c^{2} d^{2}}}\)
- \(\sqrt[6]{\dfrac{2 m^{9} n^{7}}{128 m^{3} n}}\)
-
- \(\sqrt{\dfrac{50 r^{5} s^{2}}{128 r^{2} s^{6}}}\)
- \(\sqrt[3]{\dfrac{24 m^{9} n^{7}}{375 m^{4} n}}\)
- \(\sqrt[4]{\dfrac{81 m^{2} n^{8}}{256 m^{1} n^{2}}}\)
-
- \(\dfrac{\sqrt{45 p^{9}}}{\sqrt{5 q^{2}}}\)
- \(\dfrac{\sqrt[4]{64}}{\sqrt[4]{2}}\)
- \(\dfrac{\sqrt[5]{128 x^{8}}}{\sqrt[5]{2 x^{2}}}\)
-
- \(\dfrac{\sqrt{80 q^{5}}}{\sqrt{5 q}}\)
- \(\dfrac{\sqrt[3]{-625}}{\sqrt[3]{5}}\)
- \(\dfrac{\sqrt[4]{80 m^{7}}}{\sqrt[4]{5 m}}\)
-
- \(\dfrac{\sqrt{50 m^{7}}}{\sqrt{2 m}}\)
- \(\sqrt[3]{\dfrac{1250}{2}}\)
- \(\sqrt[4]{\dfrac{486 y^{9}}{2 y^{3}}}\)
-
- \(\dfrac{\sqrt{72 n^{11}}}{\sqrt{2 n}}\)
- \(\sqrt[3]{\dfrac{162}{6}}\)
- \(\sqrt[4]{\dfrac{160 r^{10}}{5 r^{3}}}\)
- Answer
-
1.
- \(\dfrac{3}{4}\)
- \(\dfrac{2}{3}\)
- \(\dfrac{1}{3}\)
3.
- \(\dfrac{5}{3}\)
- \(\dfrac{3}{5}\)
- \(\dfrac{1}{4}\)
5.
- \(x^{2}\)
- \(p^{3}\)
- \(|q|\)
7.
- \(\dfrac{1}{y^{2}}\)
- \(u^{2}\)
- \(|v^{3}|\)
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.
- \(\dfrac{5 r^{4} \sqrt{3 r}}{s^{4}}\)
- \(\dfrac{3 a^{2} \sqrt[3]{2 a^{2}}}{|b|}\)
- \(\dfrac{2|c| \sqrt[4]{4 c}}{|d|}\)
19.
- \(\dfrac{2\left|p^{3}\right| \sqrt{7 p}}{|q|}\)
- \(\dfrac{3 s^{2} \sqrt[3]{3 s^{2}}}{t}\)
- \(\dfrac{2\left|p^{3}\right| \sqrt[4]{4 p^{3}}}{\left|q^{3}\right|}\)
21.
- \(\dfrac{4|x y|}{3}\)
- \(\dfrac{y^{2} \sqrt[3]{x}}{2}\)
- \(\dfrac{|a b| \sqrt[4]{a}}{4}\)
23.
- \(\dfrac{1}{2|p q|}\)
- \(\dfrac{2 c d \sqrt[5]{2 d^{2}}}{5}\)
- \(\dfrac{|m n| \sqrt[6]{2}}{2}\)
25.
- \(\dfrac{3 p^{4} \sqrt{p}}{|q|}\)
- \(2 \sqrt[4]{2}\)
- \(2 x \sqrt[5]{2 x}\)
27.
- \(5\left|m^{3}\right|\)
- \(5 \sqrt[3]{5}\)
- \(3|y| \sqrt[4]{3 y^{2}}\)
- Explain why \(\sqrt{x^{4}}=x^{2}\). Then explain why \(\sqrt{x^{16}}=x^{8}\).
- Explain why \(7+\sqrt{9}\) is not equal to \(\sqrt{7+9}\).
- Explain how you know that \(\sqrt[5]{x^{10}}=x^{2}\).
- 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.
b. After reviewing this checklist, what will you do to become confident for all objectives?