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Section11.10Additional Practice with Radicals and Rational Exponents

Subsection11.10.1Exercises

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 \(\{-1,\frac{1}{3}\}\text{.}\)