ExercisesExercises

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

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

25.

$\sqrt{64}$

Solution

$\sqrt{64}=4$

26.

$-\sqrt{625}$

Solution

$-\sqrt{625}=-5$

27.

$-\sqrt{-243}$

Solution

$-\sqrt{-243}=3$

28.

$3\sqrt{64}$

Solution

$3\sqrt{64}=6$

29.

$2\sqrt{-343}$

Solution

$2\sqrt{-343}=-14$

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

$x^{6/17}=\sqrt{x^6}$

31.

$x^{5/2}$

Solution

$x^{5/2}=\sqrt{x^5}$

32.

$y^{1/7}$

Solution

$y^{1/7}=\sqrt{y}$

33.

$\sqrt{y^5}$

Solution

$\sqrt{y^5}=y^{5/8}$

34.

$\sqrt{w^{24}}$

Solution

$\sqrt{w^{24}}=w^{12}$

35.

$\sqrt{t^{21}}$

Solution

$\sqrt{t^{21}}=t^{3/2}$

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.

$\sqrt{x^{10}}$

Solution

$\sqrt{x^{10}}=\sqrt{x^5}$

41.

$\sqrt{x}\sqrt{x}$

Solution

$\sqrt{x}\sqrt{x}=\sqrt{x^2}$

42.

$\sqrt{x^6y^7}$

Solution

$\sqrt{x^6y^7}=x^2y^2\sqrt{y}$

43.

$\sqrt{\sqrt{y^{24}}}$

Solution

$\sqrt{\sqrt{y^{24}}}=y^3$

Determine the solution set to each equation.

44.

$\sqrt{2x+1}=3$

Solution

The solution set is $\{4\}\text{.}$

45.

$2\sqrt{1-\frac{t}{8}}=3$

Solution

The solution set is $\{-19\}\text{.}$

46.

$\sqrt{3-x}=-4$

Solution

The solution set is $\emptyset\text{.}$

47.

$\sqrt{6w+4}=2$

Solution

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

48.

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

Solution

The solution set is $\{13\}\text{.}$

49.

$\sqrt{2x+7}+11=5-\sqrt{2x+7}$

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 $\{-1,\frac{1}{3}\}\text{.}$