### 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.

$\sqrt[3]{64}$

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