10.5E: Exercises
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Practice Makes Perfect
Exercise A: add and subtract radical expressions
In the following exercises, simplify. Assume all variables are greater than or equal to zero so that absolute values are not needed.
- a. \(8 \sqrt{2}-5 \sqrt{2}\quad\) b. \(5 \sqrt[3]{m}+2 \sqrt[3]{m}\quad\) c. \(8 \sqrt[4]{m}-2 \sqrt[4]{n}\)
- a. \(7 \sqrt{2}-3 \sqrt{2}\quad\) b. \(7 \sqrt[3]{p}+2 \sqrt[3]{p}\quad\) c. \(5 \sqrt[3]{x}-3 \sqrt[3]{x}\)
- a. \(3 \sqrt{5}+6 \sqrt{5}\quad\) b. \(9 \sqrt[3]{a}+3 \sqrt[3]{a}\quad\) c. \(5 \sqrt[4]{2 z}+\sqrt[4]{2 z}\)
- a. \(4 \sqrt{5}+8 \sqrt{5} \quad \) b. \(\sqrt[3]{m}-4 \sqrt[3]{m} \quad \) c. \(\sqrt{n}+3 \sqrt{n}\)
- a. \(3 \sqrt{2 a}-4 \sqrt{2 a}+5 \sqrt{2 a} \quad \) b. \(5 \sqrt[4]{3 a b}-3 \sqrt[4]{3 a b}-2 \sqrt[4]{3 a b}\)
- a. \(\sqrt{11 b}-5 \sqrt{11 b}+3 \sqrt{11 b} \quad \) b. \(8 \sqrt[4]{11 c d}+5 \sqrt[4]{11 c d}-9 \sqrt[4]{11 c d}\)
- a. \(8 \sqrt{3 c}+2 \sqrt{3 c}-9 \sqrt{3 c} \quad \) b. \(2 \sqrt[3]{4 p q}-5 \sqrt[3]{4 p q}+4 \sqrt[3]{4 p q}\)
- a. \(3 \sqrt{5 d}+8 \sqrt{5 d}-11 \sqrt{5 d} \quad \) b. \(11 \sqrt[3]{2 r s}-9 \sqrt[3]{2 r s}+3 \sqrt[3]{2 r s}\)
- a. \(\sqrt{27}-\sqrt{75} \quad \) b. \(\sqrt[3]{40}-\sqrt[3]{320} \quad \) c. \(\frac{1}{2} \sqrt[4]{32}+\frac{2}{3} \sqrt[4]{162}\)
- a. \(\sqrt{72}-\sqrt{98} \quad \) b. \(\sqrt[3]{24}+\sqrt[3]{81} \quad \) c. \(\frac{1}{2} \sqrt[4]{80}-\frac{2}{3} \sqrt[4]{405}\)
- a. \(\sqrt{48}+\sqrt{27} \quad \) b. \(\sqrt[3]{54}+\sqrt[3]{128} \quad \) c. \(6 \sqrt[4]{5}-\frac{3}{2} \sqrt[4]{320}\)
- a. \(\sqrt{45}+\sqrt{80} \quad \) b. \(\sqrt[3]{81}-\sqrt[3]{192} \quad \) c. \(\frac{5}{2} \sqrt[4]{80}+\frac{7}{3} \sqrt[4]{405}\)
- a. \(\sqrt{72 a^{5}}-\sqrt{50 a^{5}} \quad \) b. \(9 \sqrt[4]{80 p^{4}}-6 \sqrt[4]{405 p^{4}}\)
- a. \(\sqrt{48 b^{5}}-\sqrt{75 b^{5}} \quad \) b. \(8 \sqrt[3]{64 q^{6}}-3 \sqrt[3]{125 q^{6}}\)
- a. \(\sqrt{80 c^{7}}-\sqrt{20 c^{7}} \quad \) b. \(2 \sqrt[4]{162 r^{10}}+4 \sqrt[4]{32 r^{10}}\)
- a. \(\sqrt{96 d^{9}}-\sqrt{24 d^{9}} \quad \) b. \(5 \sqrt[4]{243 s^{6}}+2 \sqrt[4]{3 s^{6}}\)
- \(3 \sqrt{128 y^{2}}+4 y \sqrt{162}-8 \sqrt{98 y^{2}}\)
- \(3 \sqrt{75 y^{2}}+8 y \sqrt{48}-\sqrt{300 y^{2}}\)
- Answer
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1. a. \(3 \sqrt{2}\) b. \(7 \sqrt[3]{m}\) c. \(6 \sqrt[4]{m}\)
3. a. \(9 \sqrt{5}\) b. \(12 \sqrt[3]{a}\) c. \(6 \sqrt[4]{2 z}\)
5. a. \(4 \sqrt{2 a}\) b. \(0\)
7. a. \( \sqrt{3c}\) b. \(\sqrt[3]{4 p q}\)
9. a. \(-2 \sqrt{3}\) b. \(-2 \sqrt[3]{5}\) c. \(3 \sqrt[4]{2}\)
11. a. \(7 \sqrt{3}\) b. \(7 \sqrt[3]{2}\) c. \(3 \sqrt[4]{5}\)
13. a. \(a^{2} \sqrt{2 a}\) b. \(0\)
15. a. \(2 c^{3} \sqrt{5 c}\) b. \(14 r^{2} \sqrt[4]{2 r^{2}}\)
17. \(4 y \sqrt{2}\)
Exercise B: multiply radical expressions
In the following exercises, simplify.
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- \((-2 \sqrt{3})(3 \sqrt{18})\)
- \((8 \sqrt[3]{4})(-4 \sqrt[3]{18})\)
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- \((-4 \sqrt{5})(5 \sqrt{10})\)
- \((-2 \sqrt[3]{9})(7 \sqrt[3]{9})\)
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- \((5 \sqrt{6})(-\sqrt{12})\)
- \((-2 \sqrt[4]{18})(-\sqrt[4]{9})\)
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- \((-2 \sqrt{7})(-2 \sqrt{14})\)
- \((-3 \sqrt[4]{8})(-5 \sqrt[4]{6})\)
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- \(\left(4 \sqrt{12 z^{3}}\right)(3 \sqrt{9 z})\)
- \(\left(5 \sqrt[3]{3 x^{3}}\right)\left(3 \sqrt[3]{18 x^{3}}\right)\)
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- \(\left(3 \sqrt{2 x^{3}}\right)\left(7 \sqrt{18 x^{2}}\right)\)
- \(\left(-6 \sqrt[3]{20 a^{2}}\right)\left(-2 \sqrt[3]{16 a^{3}}\right)\)
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- \(\left(-2 \sqrt{7 z^{3}}\right)\left(3 \sqrt{14 z^{8}}\right)\)
- \(\left(2 \sqrt[4]{8 y^{2}}\right)\left(-2 \sqrt[4]{12 y^{3}}\right)\)
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- \(\left(4 \sqrt{2 k^{5}}\right)\left(-3 \sqrt{32 k^{6}}\right)\)
- \(\left(-\sqrt[4]{6 b^{3}}\right)\left(3 \sqrt[4]{8 b^{3}}\right)\)
- Answer
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1.
- \(-18 \sqrt{6}\)
- \(-64 \sqrt[3]{9}\)
3.
- \(-30 \sqrt{2}\)
- \(6 \sqrt[4]{2}\)
5.
- \(72 z^{2} \sqrt{3}\)
- \(45 x^{2} \sqrt[3]{2}\)
7.
- \(-42 z^{5} \sqrt{2 z}\)
- \(-8 y \sqrt[4]{6 y}\)
Exercise C: use polynomial multiplication to multiply radical expressions
In the following exercises, multiply.
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- \(\sqrt{7}(5+2 \sqrt{7})\)
- \(\sqrt[3]{6}(4+\sqrt[3]{18})\)
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- \(\sqrt{11}(8+4 \sqrt{11})\)
- \(\sqrt[3]{3}(\sqrt[3]{9}+\sqrt[3]{18})\)
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- \(\sqrt{11}(-3+4 \sqrt{11})\)
- \(\sqrt[4]{3}(\sqrt[4]{54}+\sqrt[4]{18})\)
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- \(\sqrt{2}(-5+9 \sqrt{2})\)
- \(\sqrt[4]{2}(\sqrt[4]{12}+\sqrt[4]{24})\)
- \((7+\sqrt{3})(9-\sqrt{3})\)
- \((8-\sqrt{2})(3+\sqrt{2})\)
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- \((9-3 \sqrt{2})(6+4 \sqrt{2})\)
- \((\sqrt[3]{x}-3)(\sqrt[3]{x}+1)\)
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- \((3-2 \sqrt{7})(5-4 \sqrt{7})\)
- \((\sqrt[3]{x}-5)(\sqrt[3]{x}-3)\)
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- \((1+3 \sqrt{10})(5-2 \sqrt{10})\)
- \((2 \sqrt[3]{x}+6)(\sqrt[3]{x}+1)\)
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- \((7-2 \sqrt{5})(4+9 \sqrt{5})\)
- \((3 \sqrt[3]{x}+2)(\sqrt[3]{x}-2)\)
- \((\sqrt{3}+\sqrt{10})(\sqrt{3}+2 \sqrt{10})\)
- \((\sqrt{11}+\sqrt{5})(\sqrt{11}+6 \sqrt{5})\)
- \((2 \sqrt{7}-5 \sqrt{11})(4 \sqrt{7}+9 \sqrt{11})\)
- \((4 \sqrt{6}+7 \sqrt{13})(8 \sqrt{6}-3 \sqrt{13})\)
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- \((3+\sqrt{5})^{2}\)
- \((2-5 \sqrt{3})^{2}\)
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- \((4+\sqrt{11})^{2}\)
- \((3-2 \sqrt{5})^{2}\)
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- \((9-\sqrt{6})^{2}\)
- \((10+3 \sqrt{7})^{2}\)
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- \((5-\sqrt{10})^{2}\)
- \((8+3 \sqrt{2})^{2}\)
- \((4+\sqrt{2})(4-\sqrt{2})\)
- \((7+\sqrt{10})(7-\sqrt{10})\)
- \((4+9 \sqrt{3})(4-9 \sqrt{3})\)
- \((1+8 \sqrt{2})(1-8 \sqrt{2})\)
- \((12-5 \sqrt{5})(12+5 \sqrt{5})\)
- \((9-4 \sqrt{3})(9+4 \sqrt{3})\)
- \((\sqrt[3]{3 x}+2)(\sqrt[3]{3 x}-2)\)
- \((\sqrt[3]{4 x}+3)(\sqrt[3]{4 x}-3)\)
- Answer
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1.
- \(14+5 \sqrt{7}\)
- \(4 \sqrt[3]{6}+3 \sqrt[3]{4}\)
3.
- \(44-3 \sqrt{11}\)
- \(3 \sqrt[4]{2}+\sqrt[4]{54}\)
5. \(60+2 \sqrt{3}\)
7.
- \(30+18 \sqrt{2}\)
- \(\sqrt[3]{x^{2}}-2 \sqrt[3]{x}-3\)
9.
- \(-54+13 \sqrt{10}\)
- \(2 \sqrt[3]{x^{2}}+8 \sqrt[3]{x}+6\)
11. \(23+3 \sqrt{30}\)
13. \(-439-2 \sqrt{77}\)
15.
- \(14+6 \sqrt{5}\)
- \(79-20 \sqrt{3}\)
17.
- \(87-18 \sqrt{6}\)
- \(163+60 \sqrt{7}\)
19. \(14\)
21. \(-227\)
23. \(19\)
25. \(\sqrt[3]{9 x^{2}}-4\)
Exercise D: mixed practice
- \(\frac{2}{3} \sqrt{27}+\frac{3}{4} \sqrt{48}\)
- \(\sqrt{175 k^{4}}-\sqrt{63 k^{4}}\)
- \(\frac{5}{6} \sqrt{162}+\frac{3}{16} \sqrt{128}\)
- \(\sqrt[3]{24}+\sqrt[3]{ 81}\)
- \(\frac{1}{2} \sqrt[4]{80}-\frac{2}{3} \sqrt[4]{405}\)
- \(8 \sqrt[4]{13}-4 \sqrt[4]{13}-3 \sqrt[4]{13}\)
- \(5 \sqrt{12 c^{4}}-3 \sqrt{27 c^{6}}\)
- \(\sqrt{80 a^{5}}-\sqrt{45 a^{5}}\)
- \(\frac{3}{5} \sqrt{75}-\frac{1}{4} \sqrt{48}\)
- \(21 \sqrt[3]{9}-2 \sqrt[3]{9}\)
- \(8 \sqrt[3]{64 q^{6}}-3 \sqrt[3]{125 q^{6}}\)
- \(11 \sqrt{11}-10 \sqrt{11}\)
- \(\sqrt{3} \cdot \sqrt{21}\)
- \((4 \sqrt{6})(-\sqrt{18})\)
- \((7 \sqrt[3]{4})(-3 \sqrt[3]{18})\)
- \(\left(4 \sqrt{12 x^{5}}\right)\left(2 \sqrt{6 x^{3}}\right)\)
- \((\sqrt{29})^{2}\)
- \((-4 \sqrt{17})(-3 \sqrt{17})\)
- \((-4+\sqrt{17})(-3+\sqrt{17})\)
- \(\left(3 \sqrt[4]{8 a^{2}}\right)\left(\sqrt[4]{12 a^{3}}\right)\)
- \((6-3 \sqrt{2})^{2}\)
- \(\sqrt{3}(4-3 \sqrt{3})\)
- \(\sqrt[3]{3}(2 \sqrt[3]{9}+\sqrt[3]{18})\)
- \((\sqrt{6}+\sqrt{3})(\sqrt{6}+6 \sqrt{3})\)
- Answer
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1. \(5\sqrt{3}\)
3. \(9\sqrt{2}\)
5. \(-\sqrt[4]{5}\)
7. \(10 c^{2} \sqrt{3}-9 c^{3} \sqrt{3}\)
9. \(2 \sqrt{3}\)
11. \(17 q^{2}\)
13. \(3 \sqrt{7}\)
15. \(-42 \sqrt[3]{9}\)
17. \(29\)
19. \(29-7 \sqrt{17}\)
21. \(72-36 \sqrt{2}\)
23. \(6+3 \sqrt[3]{2}\)
Exercise E: writing exercises
- Explain when a radical expression is in simplest form.
- 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.
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- Explain why \((-\sqrt{n})^{2}\) is always non-negative, for \(n \geq 0\).
- Explain why \(-(\sqrt{n})^{2}\) is always non-positive, for \(n \geq 0\).
- Use the binomial square pattern to simplify \((3+\sqrt{2})^{2}\). Explain all your steps.
- Answer
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1. Answers will vary
3. Answers will vary
Self Check
a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
b. On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?