2.ED: Exercises for Radical Expressions and Equations

Simplifying Expressions with Roots

Exercise A

In the following exercises, simplify.

1. 1. \(\sqrt{225}\)
   2. \(-\sqrt{16}\)
2. 1. \(-\sqrt{169}\)
   2. \(\sqrt{-8}\)
3. 1. \(\sqrt[3]{8}\)
   2. \(\sqrt[4]{81}\)
   3. \(\sqrt[5]{243}\)
4. 1. \(\sqrt[3]{-512}\)
   2. \(\sqrt[4]{-81}\)
   3. \(\sqrt[5]{-1}\)

Answer

1.

1. \(15\)
2. \(-4\)

3.

1. \(2\)
2. \(3\)
3. \(3\)

Exercise B

In the following exercises, estimate each root between two consecutive whole numbers.

1. 1. \(\sqrt{68}\)
   2. \(\sqrt[3]{84}\)

Answer

1.

1. \(8<\sqrt{68}<9\)
2. \(4<\sqrt[3]{84}<5\)

Exercise C

In the following exercises, approximate each root and round to two decimal places.

1. 1. \(\sqrt{37}\)
   2. \(\sqrt[3]{84}\)
   3. \(\sqrt[4]{125}\)

Answer

1. Solve for yourself

Exercise D

In the following exercises, simplify using absolute values as necessary.

1. 1. \(\sqrt[3]{a^{3}}\)
   2. \(\sqrt[7]{b^{7}}\)
2. 1. \(\sqrt{a^{14}}\)
   2. \(\sqrt{w^{24}}\)
3. 1. \(\sqrt[4]{m^{8}}\)
   2. \(\sqrt[5]{n^{20}}\)
4. 1. \(\sqrt{121 m^{20}}\)
   2. \(-\sqrt{64 a^{2}}\)
5. 1. \(\sqrt[3]{216 a^{6}}\)
   2. \(\sqrt[5]{32 b^{20}}\)
6. 1. \(\sqrt{144 x^{2} y^{2}}\)
   2. \(\sqrt{169 w^{8} y^{10}}\)
   3. \(\sqrt[3]{8 a^{51} b^{6}}\)

Answer

1.

1. \(a\)
2. \(|b|\)

3.

1. \(m^{2}\)
2. \(n^{4}\)

5.

1. \(6a^{2}\)
2. \(2b^{4}\)

Simplifying Radical Expressions

Exercise A

In the following exercises, use the Product Property to simplify radical expressions.

1. \(\sqrt{125}\)
2. \(\sqrt{675}\)
3. 1. \(\sqrt[3]{625}\)
   2. \(\sqrt[6]{128}\)

Answer

1. \(5\sqrt{5}\)

3.

1. \(5 \sqrt[3]{5}\)
2. \(2 \sqrt[6]{2}\)

Exercise B

In the following exercises, simplify using absolute value signs as needed.

1. 1. \(\sqrt{a^{23}}\)
   2. \(\sqrt[3]{b^{8}}\)
   3. \(\sqrt[8]{c^{13}}\)
2. 1. \(\sqrt{80 s^{15}}\)
   2. \(\sqrt[5]{96 a^{7}}\)
   3. \(\sqrt[6]{128 b^{7}}\)
3. 1. \(\sqrt{96 r^{3} s^{3}}\)
   2. \(\sqrt[3]{80 x^{7} y^{6}}\)
   3. \(\sqrt[4]{80 x^{8} y^{9}}\)
4. 1. \(\sqrt[5]{-32}\)
   2. \(\sqrt[8]{-1}\)
5. 1. \(8+\sqrt{96}\)
   2. \(\frac{2+\sqrt{40}}{2}\)

Answer

2.

1. \(4\left|s^{7}\right| \sqrt{5 s}\)
2. \(2 a \sqrt[5]{3 a^{2}}\)
3. \(2|b| \sqrt[6]{2 b}\)

4.

1. \(-2\)
2. not real

Exercise C

In the following exercises, use the Quotient Property to simplify square roots.

1. 1. \(\sqrt{\frac{72}{98}}\)
   2. \(\sqrt[3]{\frac{24}{81}}\)
   3. \(\sqrt[4]{\frac{6}{96}}\)
2. 1. \(\sqrt{\frac{y^{4}}{y^{8}}}\)
   2. \(\sqrt[5]{\frac{u^{21}}{u^{11}}}\)
   3. \(\sqrt[6]{\frac{v^{30}}{v^{12}}}\)
3. \(\sqrt{\frac{300 m^{5}}{64}}\)
4. 1. \(\sqrt{\frac{28 p^{7}}{q^{2}}}\)
   2. \(\sqrt[3]{\frac{81 s^{8}}{t^{3}}}\)
   3. \(\sqrt[4]{\frac{64 p^{15}}{q^{12}}}\)
5. 1. \(\sqrt{\frac{27 p^{2} q}{108 p^{4} q^{3}}}\)
   2. \(\sqrt[3]{\frac{16 c^{5} d^{7}}{250 c^{2} d^{2}}}\)
   3. \(\sqrt[6]{\frac{2 m^{9} n^{7}}{128 m^{3} n}}\)
6. 1. \(\frac{\sqrt{80 q^{5}}}{\sqrt{5 q}}\)
   2. \(\frac{\sqrt[3]{-625}}{\sqrt[3]{5}}\)
   3. \(\frac{\sqrt[4]{80 m^{7}}}{\sqrt[4]{5 m}}\)

Answer

1.

1. \(\frac{6}{7}\)
2. \(\frac{2}{3}\)
3. \(\frac{1}{2}\)

3. \(\frac{10 m^{2} \sqrt{3 m}}{8}\)

5.

1. \(\frac{1}{2|p q|}\)
2. \(\frac{2 c d \sqrt[5]{2 d^{2}}}{5}\)
3. \(\frac{|m n| \sqrt[6]{2}}{2}\)

Simplifying Rational Exponents

Exercise A

In the following exercises, write as a radical expression.

1. 1. \(r^{\frac{1}{2}}\)
   2. \(s^{\frac{1}{3}}\)
   3. \(t^{\frac{1}{4}}\)

Answer

1.

1. \(\sqrt{r}\)
2. \(\sqrt[3]{s}\)
3. \(\sqrt[4]{t}\)

Exercise B

In the following exercises, write with a rational exponent.

1. 1. \(\sqrt{21p}\)
   2. \(\sqrt[4]{8q}\)
   3. \(4\sqrt[6]{36r}\)

Answer

1. Solve for yourself

Exercise C

In the following exercises, simplify.

1. 1. \(625^{\frac{1}{4}}\)
   2. \(243^{\frac{1}{5}}\)
   3. \(32^{\frac{1}{5}}\)
2. 1. \((-1,000)^{\frac{1}{3}}\)
   2. \(-1,000^{\frac{1}{3}}\)
   3. \((1,000)^{-\frac{1}{3}}\)
3. 1. \((-32)^{\frac{1}{5}}\)
   2. \((243)^{-\frac{1}{5}}\)
   3. \(-125^{\frac{1}{3}}\)

Answer

1.

1. \(5\)
2. \(3\)
3. \(2\)

3.

1. \(-2\)
2. \(\frac{1}{3}\)
3. \(-5\)

Exercise D

In the following exercises, write with a rational exponent.

1. 1. \(\sqrt[4]{r^{7}}\)
   2. \((\sqrt[5]{2 p q})^{3}\)
   3. \(\sqrt[4]{\left(\frac{12 m}{7 n}\right)^{3}}\)

Answer

1. Solve for yourself

Exercise E

In the following exercises, simplify.

1. 1. \(25^{\frac{3}{2}}\)
   2. \(9^{-\frac{3}{2}}\)
   3. \((-64)^{\frac{2}{3}}\)
2. 1. \(-64^{\frac{3}{2}}\)
   2. \(-64^{-\frac{3}{2}}\)
   3. \((-64)^{\frac{3}{2}}\)

Answer

1.

1. \(125\)
2. \(\frac{1}{27}\)
3. \(16\)

Exercise F

In the following exercises, simplify.

1. 1. \(6^{\frac{5}{2}} \cdot 6^{\frac{1}{2}}\)
   2. \(\left(b^{15}\right)^{\frac{3}{5}}\)
   3. \(\frac{w^{\frac{2}{7}}}{w^{\frac{9}{7}}}\)
2. 1. \(\frac{a^{\frac{3}{4}} \cdot a^{-\frac{1}{4}}}{a^{-\frac{10}{4}}}\)
   2. \(\left(\frac{27 b^{\frac{2}{3}} c^{-\frac{5}{2}}}{b^{-\frac{7}{3}} c^{\frac{1}{2}}}\right)^{\frac{1}{3}}\)

Answer

1.

1. \(6^{3}\)
2. \(b^{9}\)
3. \(\frac{1}{w}\)

Adding, Subtracting, and Multiplying Radical Expressions

Exercise A

In the following exercises, simplify.

1. 1. \(7 \sqrt{2}-3 \sqrt{2}\)
   2. \(7 \sqrt[3]{p}+2 \sqrt[3]{p}\)
   3. \(5 \sqrt[3]{x}-3 \sqrt[3]{x}\)
2. 1. \(\sqrt{11 b}-5 \sqrt{11 b}+3 \sqrt{11 b}\)
   2. \(8 \sqrt[4]{11 c d}+5 \sqrt[4]{11 c d}-9 \sqrt[4]{11 c d}\)
3. 1. \(\sqrt{48}+\sqrt{27}\)
   2. \(\sqrt[3]{54}+\sqrt[3]{128}\)
   3. \(6 \sqrt[4]{5}-\frac{3}{2} \sqrt[4]{320}\)
4. 1. \(\sqrt{80 c^{7}}-\sqrt{20 c^{7}}\)
   2. \(2 \sqrt[4]{162 r^{10}}+4 \sqrt[4]{32 r^{10}}\)
5. \(3 \sqrt{75 y^{2}}+8 y \sqrt{48}-\sqrt{300 y^{2}}\)

Answer

1.

1. \(4\sqrt{2}\)
2. \(9\sqrt[3]{p}\)
3. \(2\sqrt[3]{x}\)

3.

1. \(7\sqrt{3}\)
2. \(7\sqrt[3]{2}\)
3. \(3\sqrt[4]{5}\)

5. \(37 y \sqrt{3}\)

Exercise B

In the following exercises, simplify.

1. 1. \((5 \sqrt{6})(-\sqrt{12})\)
   2. \((-2 \sqrt[4]{18})(-\sqrt[4]{9})\)
2. 1. \(\left(3 \sqrt{2 x^{3}}\right)\left(7 \sqrt{18 x^{2}}\right)\)
   2. \(\left(-6 \sqrt[3]{20 a^{2}}\right)\left(-2 \sqrt[3]{16 a^{3}}\right)\)

Answer

2.

1. \(126 x^{2} \sqrt{2}\)
2. \(48 a \sqrt[3]{a^{2}}\)

Exercise C

In the following exercises, multiply.

1. 1. \(\sqrt{11}(8+4 \sqrt{11})\)
   2. \(\sqrt[3]{3}(\sqrt[3]{9}+\sqrt[3]{18})\)
2. 1. \((3-2 \sqrt{7})(5-4 \sqrt{7})\)
   2. \((\sqrt[3]{x}-5)(\sqrt[3]{x}-3)\)
3. \((2 \sqrt{7}-5 \sqrt{11})(4 \sqrt{7}+9 \sqrt{11})\)
4. 1. \((4+\sqrt{11})^{2}\)
   2. \((3-2 \sqrt{5})^{2}\)
5. \((7+\sqrt{10})(7-\sqrt{10})\)
6. \((\sqrt[3]{3 x}+2)(\sqrt[3]{3 x}-2)\)

Answer

2.

1. \(71-22 \sqrt{7}\)
2. \(\sqrt[3]{x^{2}}-8 \sqrt[3]{x}+15\)

4.

1. \(27+8 \sqrt{11}\)
2. \(29-12 \sqrt{5}\)

6. \(\sqrt[3]{9 x^{2}}-4\)

Dividing Radical Expressions

Exercise A

In the following exercises, simplify.

1. 1. \(\frac{\sqrt{48}}{\sqrt{75}}\)
   2. \(\frac{\sqrt[3]{81}}{\sqrt[3]{24}}\)
2. 1. \(\frac{\sqrt{320 m n^{-5}}}{\sqrt{45 m^{-7} n^{3}}}\)
   2. \(\frac{\sqrt[3]{16 x^{4} y^{-2}}}{\sqrt[3]{-54 x^{-2} y^{4}}}\)

Answer

2.

1. \(\frac{8 m^{4}}{3 n^{4}}\)
2. \(-\frac{x^{2}}{2 y^{2}}\)

Exercise B

In the following exercises, rationalize the denominator.

1. 1. \(\frac{8}{\sqrt{3}}\)
   2. \(\sqrt{\frac{7}{40}}\)
   3. \(\frac{8}{\sqrt{2 y}}\)
2. 1. \(\frac{1}{\sqrt[3]{11}}\)
   2. \(\sqrt[3]{\frac{7}{54}}\)
   3. \(\frac{3}{\sqrt[3]{3 x^{2}}}\)
3. 1. \(\frac{1}{\sqrt[4]{4}}\)
   2. \(\sqrt[4]{\frac{9}{32}}\)
   3. \(\frac{6}{\sqrt[4]{9 x^{3}}}\)

Answer

2.

1. \(\frac{\sqrt[3]{121}}{11}\)
2. \(\frac{\sqrt[3]{28}}{6}\)
3. \(\frac{\sqrt[3]{9 x}}{x}\)

Exercise C

In the following exercises, simplify.

1. \(\frac{7}{2-\sqrt{6}}\)
2. \(\frac{\sqrt{5}}{\sqrt{n}-\sqrt{7}}\)
3. \(\frac{\sqrt{x}+\sqrt{8}}{\sqrt{x}-\sqrt{8}}\)

Answer

1. \(-\frac{7(2+\sqrt{6})}{2}\)

3. \(\frac{(\sqrt{x}+2 \sqrt{2})^{2}}{x-8}\)

Solving Radical Equations

Exercise A

In the following exercises, solve.

1. \(\sqrt{4 x-3}=7\)
2. \(\sqrt{5 x+1}=-3\)
3. \(\sqrt[3]{4 x-1}=3\)
4. \(\sqrt{u-3}+3=u\)
5. \(\sqrt[3]{4 x+5}-2=-5\)
6. \((8 x+5)^{\frac{1}{3}}+2=-1\)
7. \(\sqrt{y+4}-y+2=0\)
8. \(2 \sqrt{8 r+1}-8=2\)

Answer

2. no solution

4. \(u=3, u=4\)

6. \(x=-4\)

8. \(r=3\)

Exercise B

In the following exercises, solve.

1. \(\sqrt{10+2 c}=\sqrt{4 c+16}\)
2. \(\sqrt[3]{2 x^{2}+9 x-18}=\sqrt[3]{x^{2}+3 x-2}\)
3. \(\sqrt{r}+6=\sqrt{r+8}\)
4. \(\sqrt{x+1}-\sqrt{x-2}=1\)

Answer

2. \(x=-8, x=2\)

4. \(x=3\)

Exercise C

In the following exercises, solve. Round approximations to one decimal place.

1. Landscaping Reed wants to have a square garden plot in his backyard. He has enough compost to cover an area of \(75\) square feet. Use the formula \(s=\sqrt{A}\) to find the length of each side of his garden. Round your answers to th nearest tenth of a foot.
2. Accident investigation An accident investigator measured the skid marks of one of the vehicles involved in an accident. The length of the skid marks was \(175\) feet. Use the formula \(s=\sqrt{24d}\) to find the speed of the vehicle before the brakes were applied. Round your answer to the nearest tenth.

Answer

2. \(64.8\) feet

Complex Numbers

Exercise A

In the following exercises, write each expression in terms of \(i\) and simplify if possible.

1. 1. \(\sqrt{-100}\)
   2. \(\sqrt{-13}\)
   3. \(\sqrt{-45}\)

Answer

Solve for yourself

Exercise B

In the following exercises, add or subtract.

1. \(\sqrt{-50}+\sqrt{-18}\)
2. \((8-i)+(6+3 i)\)
3. \((6+i)-(-2-4 i)\)
4. \((-7-\sqrt{-50})-(-32-\sqrt{-18})\)

Answer

1. \(8 \sqrt{2} i\)

3. \(8+5 i\)

Exercise C

In the following exercises, multiply.

1. \((-2-5 i)(-4+3 i)\)
2. \(-6 i(-3-2 i)\)
3. \(\sqrt{-4} \cdot \sqrt{-16}\)
4. \((5-\sqrt{-12})(-3+\sqrt{-75})\)

Answer

1. \(23+14 i\)

3. \(-6\)

Exercise D

In the following exercises, multiply using the Product of Binomial Squares Pattern.

1. \((-2-3 i)^{2}\)

Answer

1. \(-5-12 i\)

Exercise E

In the following exercises, multiply using the Product of Complex Conjugates Pattern.

1. \((9-2 i)(9+2 i)\)

Answer

Solve for yourself

Exercise F

In the following exercises, divide.

1. \(\frac{2+i}{3-4 i}\)
2. \(\frac{-4}{3-2 i}\)

Answer

1. \(\frac{2}{25}+\frac{11}{25} i\)

Exercise G

In the following exercises, simplify.

1. \(i^{48}\)
2. \(i^{255}\)

Answer

1. \(1\)

2. \(-i\)