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Section 12.10 Additional Practice with Radicals and Rational Exponents

Exercises Exercises

Simplify each expression. Please note that this includes factoring out perfect squares, rationalizing denominators, and combining like terms.

1.

\(\sqrt{28}\)

Solution

\(\sqrt{28}=2\sqrt{7}\)

2.

\(\sqrt{98}\)

Solution

\(\sqrt{98}=7\sqrt{2}\)

3.

\(-\sqrt{800}\)

Solution

\(-\sqrt{800}=-20\sqrt{2}\)

4.

\(5\sqrt{125}\)

Solution

\(5\sqrt{125}=25\sqrt{5}\)

5.

\(\frac{\sqrt{32}}{8}\)

Solution

\(\frac{\sqrt{32}}{8}=\frac{\sqrt{2}}{2}\)

6.

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

Solution

\(\frac{6}{\sqrt{3}}=2\sqrt{3}\)

7.

\(\frac{4}{\sqrt{8}}\)

Solution

\(\frac{4}{\sqrt{8}}=\sqrt{2}\)

8.

\(-\frac{2}{\sqrt{48}}\)

Solution

\(-\frac{2}{\sqrt{48}}=-\frac{\sqrt{3}}{6}\)

9.

\(-\frac{7}{3\sqrt{63}}\)

Solution

\(-\frac{7}{3\sqrt{63}}=-\frac{\sqrt{7}}{9}\)

10.

\(\frac{3}{25\sqrt{200}}\)

Solution

\(\frac{3}{25\sqrt{200}}=\frac{3\sqrt{2}}{500}\)

11.

\(\frac{12}{\sqrt{288}}\)

Solution

\(\frac{12}{\sqrt{288}}=\frac{\sqrt{2}}{2}\)

12.

\(2\sqrt{8}-\sqrt{32}\)

Solution

\(2\sqrt{8}-\sqrt{32}=0\)

13.

\(\sqrt{80}+3\sqrt{20}\)

Solution

\(\sqrt{80}+3\sqrt{20}=10\sqrt{5}\)

14.

\(3\sqrt{18}-2\sqrt{24}\)

Solution

\(3\sqrt{18}-2\sqrt{24}=9\sqrt{2}-4\sqrt{6}\)

15.

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

Solution

\((2+\sqrt{7})^2=11+4\sqrt{7}\)

16.

\((6-\sqrt{20})(6+\sqrt{20})\)

Solution

\((6-\sqrt{20})(6+\sqrt{20})=16\)

17.

\((\sqrt{8}+3)(\sqrt{8}-3)\)

Solution

\((\sqrt{8}+3)(\sqrt{8}-3)=-1\)

18.

\((1-\sqrt{72})^2\)

Solution

\((1-\sqrt{72})^2=73-12\sqrt{2}\)

19.

\(\frac{6}{\sqrt{2}}-5\sqrt{18}\)

Solution

\(\frac{6}{\sqrt{2}}-5\sqrt{18}=-12\sqrt{2}\)

20.

\(\frac{2}{\sqrt{125}}+\sqrt{500}\)

Solution

\(\frac{2}{\sqrt{125}}+\sqrt{500}=\frac{252\sqrt{5}}{25}\)

21.

\(\frac{30}{\sqrt{20}}+\frac{12}{\sqrt{45}}\)

Solution

\(\frac{30}{\sqrt{20}}+\frac{12}{\sqrt{45}}=\frac{19\sqrt{5}}{5}\)

22.

\(\frac{2}{\sqrt{6}-\sqrt{2}}\)

Solution

\(\frac{2}{\sqrt{6}-\sqrt{2}}=\frac{\sqrt{6}+\sqrt{2}}{2}\)

23.

\(\frac{10}{\sqrt{6}-4}\)

Solution

\(\frac{10}{\sqrt{6}-4}=-4-\sqrt{6}\)

24.

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

Solution

\(\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}=4+\sqrt{15}\)

Simplify each radical expression.

25.

\(\sqrt[3]{64}\)

Solution

\(\sqrt[3]{64}=4\)

26.

\(-\sqrt[4]{625}\)

Solution

\(-\sqrt[4]{625}=-5\)

27.

\(-\sqrt[5]{-243}\)

Solution

\(-\sqrt[5]{-243}=3\)

28.

\(3\sqrt[6]{64}\)

Solution

\(3\sqrt[6]{64}=6\)

29.

\(2\sqrt[3]{-343}\)

Solution

\(2\sqrt[3]{-343}=-14\)

Convert each exponential expression to a radical expression and each radical expression to an exponential expression. When converting to a rational exponent, reduce the exponent if possible. Assume that all variables represent positive values.

30.

\(x^{6/17}\)

Solution

\(x^{6/17}=\sqrt[17]{x^6}\)

31.

\(x^{5/2}\)

Solution

\(x^{5/2}=\sqrt{x^5}\)

32.

\(y^{1/7}\)

Solution

\(y^{1/7}=\sqrt[7]{y}\)

33.

\(\sqrt[8]{y^5}\)

Solution

\(\sqrt[8]{y^5}=y^{5/8}\)

34.

\(\sqrt{w^{24}}\)

Solution

\(\sqrt{w^{24}}=w^{12}\)

35.

\(\sqrt[14]{t^{21}}\)

Solution

\(\sqrt[14]{t^{21}}=t^{3/2}\)

Determine the value of each expression.

36.

\(27^{4/3}\)

Solution

\(27^{4/3}=81\)

37.

\(25^{3/2}\)

Solution

\(25^{3/2}=125\)

38.

\(16^{-3/4}\)

Solution

\(16^{-3/4}=\frac{1}{8}\)

39.

\(32^{-3/5}\)

Solution

\(32^{-3/5}=\frac{1}{8}\)

Simplify each radical expression after first rewriting the expression in exponential form. Assume that all variables represent positive values.

40.

\(\sqrt[4]{x^{10}}\)

Solution

\(\sqrt[4]{x^{10}}=\sqrt{x^5}\)

41.

\(\sqrt{x}\sqrt[6]{x}\)

Solution

\(\sqrt{x}\sqrt[6]{x}=\sqrt[3]{x^2}\)

42.

\(\sqrt[3]{x^6y^7}\)

Solution

\(\sqrt[3]{x^6y^7}=x^2y^2\sqrt[3]{y}\)

43.

\(\sqrt{\sqrt[4]{y^{24}}}\)

Solution

\(\sqrt{\sqrt[4]{y^{24}}}=y^3\)

Determine the solution set to each equation.

44.

\(\sqrt{2x+1}=3\)

Solution

The solution set is \(\{4\}\text{.}\)

45.

\(2\sqrt[3]{1-\frac{t}{8}}=3\)

Solution

The solution set is \(\{-19\}\text{.}\)

46.

\(\sqrt{3-x}=-4\)

Solution

The solution set is \(\emptyset\text{.}\)

47.

\(\sqrt[5]{6w+4}=2\)

Solution

The solution set is \(\{\frac{14}{3}\}\text{.}\)

48.

\(3\sqrt{y-4}+5=14\)

Solution

The solution set is \(\{13\}\text{.}\)

49.

\(\sqrt[3]{2x+7}+11=5-\sqrt[3]{2x+7}\)

Solution

The solution set is \(\{-17\}\text{.}\)

50.

\(-\frac{\sqrt{8-t}}{3}+6=1\)

Solution

The solution set is \(\{-217\}\text{.}\)

51.

\(x=3+\sqrt{x-1}\)

Solution

The solution set is \(\{5\}\text{.}\)

52.

\(\sqrt{7-x}-4\sqrt{x+10}=0\)

Solution

The solution set is \(\{-9\}\text{.}\)

53.

\(\sqrt{3x+4}=2-\sqrt{x+2}\)

Solution

The solution set is \(\{-1\}\text{.}\)

54.

\(\sqrt{2y-5}-3\sqrt{y+1}=-7\)

Solution

The solution set is \(\{15\}\text{.}\)

55.

\(\sqrt{6t+7}-\sqrt{3t+3}=1\)

Solution

The solution set is \(\{-1,\frac{1}{3}\}\text{.}\)