Skip to main content
$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 8.E: Review Exercises and Sample Exam

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

## Review Exercises

(Assume all variables represent nonnegative numbers.)

Exercise $$\PageIndex{1}$$ Radicals

Simplify.

1. $$\sqrt{36}$$
2. $$\sqrt{425}$$
3. $$\sqrt{−16}$$
4. $$-\sqrt{9}$$
5. $$\sqrt[3]{125}$$
6. $$3\sqrt[3]{−8}$$
7. $$\sqrt[3]{\frac{1}{64}}$$
8. $$−5\sqrt[3]{−27}$$
9. $$\sqrt{40}$$
10. $$−3\sqrt{50}$$
11. $$\sqrt{\frac{98}{81}}$$
12. $$\sqrt{\frac{11}{21}}$$
13. $$5\sqrt[3]{192}$$
14. $$2\sqrt[3]{−54}$$
Answer

1. 6

3. Not a real number

5. 5

7. $$\frac{1}{4}$$

9. $$2\sqrt{10}$$

11. $$\frac{7 \sqrt{2}}{9}$$

13. $$20\sqrt[3]{3}$$

Exercise $$\PageIndex{2}$$ Simplifying Radical Expressions

Simplify.

1. $$\sqrt{49x^{2}}$$
2. $$\sqrt{25a^{2}b^{2}}$$
3. $$\sqrt{75x^{3}y^{2}}$$
4. $$\sqrt{200m^{4}n^{3}}$$
5. $$\sqrt{\frac{18 x^{3}}{25 y^{2}}}$$
6. $$\sqrt{\frac{108 x^{3}}{49 y^{4}}}$$
7. $$\sqrt[3]{216 x^{3}}$$
8. $$\sqrt[3]{−125x^{6}y^{3}}$$
9. $$\sqrt[3]{27a^{7}b^{5}c^{3}}$$
10. $$\sqrt[3]{120x^{9}y^{4}}$$
Answer

1. $$7x$$

3. 5xy$$\sqrt{3x}$$

5. $$\frac{3x\sqrt{2x}}{5y}$$

7. $$6x$$

9. $$3a^{2}bc\sqrt[3]{ab^{2}}$$

Exercise $$\PageIndex{3}$$ Simplifying Radical Expressions

Use the distance formula to calculate the distance between the given two points.

1. $$(5, −8)$$ and $$(2, −10)$$
2. $$(−7, −1)$$ and $$(−6, 1)$$
3. $$(−10, −1)$$ and $$(0, −5)$$
4. $$(5, −1)$$ and $$(−2, −2)$$
Answer

1. $$\sqrt{13}$$

3. $$2\sqrt{29}$$

Exercise $$\PageIndex{4}$$ Adding and Subtracting Radical Expressions

Simplify.

1. $$8\sqrt{3}+3\sqrt{3}$$
2. $$12\sqrt{10}−2\sqrt{10}$$
3. $$14\sqrt{3}+5\sqrt{2}−5\sqrt{3}−6\sqrt{2}$$
4. $$22\sqrt{ab}−5\sqrt{ab}+7\sqrt{ab}−2\sqrt{ab}$$
5. $$7\sqrt{x}−(3\sqrt{x}+2\sqrt{y})$$
6. $$(8y\sqrt{x}−7x\sqrt{y})−(5x\sqrt{y}−12y\sqrt{x})$$
7. $$\sqrt{45}+\sqrt{12}−\sqrt{20}−\sqrt{75}$$
8. $$\sqrt{24}−\sqrt{32}+\sqrt{54}−2\sqrt{32}$$
9. $$2 \sqrt{3 x^{2}}+\sqrt{45 x}-x \sqrt{27}+\sqrt{20 x}$$
10. $$\sqrt{56a^{2}b}+\sqrt{8a^{2}b^{2}}−\sqrt{224a^{2}b}−a\sqrt{18b^{2}}$$
11. $$5y\sqrt{4x^{2}y}−(x\sqrt{16y^{3}}−2\sqrt{9x^{2}y^{3}})$$
12. $$(2b\sqrt{9a^{2}c}−3a\sqrt{16b^{2}c})−(\sqrt{64a^{2}b^{2}c}−9b\sqrt{a^{2}c})$$
13. $$\sqrt[3]{216x}−\sqrt[3]{125xy}−\sqrt[3]{8x}$$
14. $$\sqrt[3]{128x^{3}}−2x\sqrt[3]{54}+3\sqrt[3]{2x^{3}}$$
15. $$\sqrt[3]{8x^{3}y}−2x\sqrt[3]{8y}+\sqrt[3]{27x^{3}y}+x\sqrt[3]{y}$$
16. $$\sqrt[3]{27a^{3}b}−3\sqrt[3]{8ab^{3}}+a\sqrt[3]{64b}−b\sqrt[3]{a}$$
Answer

1. $$11\sqrt{3}$$

3. $$9 \sqrt{3}-\sqrt{2}$$

5. $$4 \sqrt{x}-2 \sqrt{y}$$

7.$$\sqrt{5}-3 \sqrt{3}$$

9. $$-\sqrt{3} x+5 \sqrt{5} \sqrt{x}$$

11. $$12xy\sqrt{y}$$

13. $$4 \sqrt[3]{x}-5 \sqrt[3]{x y}$$

15. $$2 x \sqrt[3]{y}$$

Exercise $$\PageIndex{5}$$ Multiplying and Dividing Radical Expressions

Multiply.

1. $$\sqrt{3}\cdot\sqrt{6}$$
2. $$(3\sqrt{5})^{2}$$
3. $$\sqrt{2}(\sqrt{3}−\sqrt{6})$$
4. $$(\sqrt{2}−\sqrt{6})^{2}$$
5. $$(1−\sqrt{5})(1+\sqrt{5})$$
6. $$(2\sqrt{3}+\sqrt{5})(3\sqrt{2}−2\sqrt{5})$$
7. $$\sqrt[3]{2a^{2}}\cdot\sqrt[3]{4a}$$
8. $$\sqrt[3]{25a^{2}b}\cdot\sqrt[3]{5a^{2}b^{2}}$$
Answer

1. $$3\sqrt{2}$$

3. $$\sqrt{6}-2 \sqrt{3}$$

5. $$−4$$

7. $$2a$$

Exercise $$\PageIndex{6}$$ Multiplying and Dividing Radical Expressions

Divide.

1. $$\frac{\sqrt{72}}{\sqrt{4}}$$
2. $$10 \frac{\sqrt{48}}{\sqrt{64}}$$
3. $$\frac{\sqrt{98 x^{4} y^{2}}}{\sqrt{36 x^{2}}}$$
4. $$\frac{\sqrt[3]{81 x^{6} y^{7}}}{\sqrt[3]{8 y^{3}}}$$
Answer

1. $$3\sqrt{2}$$

3. $$\frac{7xy \sqrt{2}}{6}$$

Exercise $$\PageIndex{7}$$ Multiplying and Dividing Radical Expressions

Rationalize the denominator.

1. $$\frac{2}{\sqrt{7}}$$
2. $$\frac{\sqrt{6}}{\sqrt{3}}$$
3. $$\sqrt{\frac{14}{2 x}}$$
4. $$\sqrt{\frac{12}{15}}$$
5. $$\sqrt[3]{\frac{1}{2 x^{2}}}$$
6. $$\sqrt[3]{\frac{5 a^{2} b^{5}}{a b^{2}}}$$
7. $$\frac{1}{\sqrt{3}-\sqrt{2}}$$
8. $$\frac{\sqrt{2}-\sqrt{6}}{\sqrt{2}+\sqrt{6}}$$
Answer

1. $$\frac{2 \sqrt{7}}{7}$$

3. $$\frac{\sqrt{7} \sqrt{x}}{x}$$

5. $$\frac{2^{\frac{2}{3}} \sqrt[3]{x}}{2 x}$$

7. $$\sqrt{3}+\sqrt{2}$$

Exercise $$\PageIndex{8}$$ Rational Exponents

Express in radical form.

1. $$7^{1/2}$$
2. $$3^{2/3}$$
3. $$x^{4/5}$$
4. $$y^{−3/4}$$
Answer

1. $$\sqrt{7}$$

3. $$\sqrt[5]{x^{4}}$$

Exercise $$\PageIndex{9}$$Rational Exponents

Write as a radical and then simplify.

1. $$4^{1/2}$$
2. $$50^{1/2}$$
3. $$4^{2/3}$$
4. $$81^{1/3}$$
5. $$(\frac{1}{4})^{3/2}$$
6. $$(\frac{12}{16})^{−1/3}$$
Answer

1. $$2$$

3. $$2\sqrt[3]{2}$$

5. $$\frac{1}{8}$$

Exercise $$\PageIndex{10}$$ Rational Exponent

Perform the operations and simplify. Leave answers in exponential form.

1. $$3^{1/2}\cdot 3^{3/2}$$
2. $$2^{1/2}\cdot 2^{1/3}$$
3. $$\frac{4^{\frac{3}{2}}}{4^{\frac{1}{2}}}$$
4. $$\frac{9^{\frac{3}{4}}}{9^{\frac{1}{4}}}$$
5. $$\left(36 x^{4} y^{2}\right)^{\frac{1}{2}}$$
6. $$(8x^{6}y^{9})^{1/3}$$
7. $$\left(\frac{a^{\frac{4}{3}}}{a^{\frac{1}{2}}}\right)^{\frac{2}{5}}$$
8. $$\left(\frac{16 x^{\frac{4}{3}}}{y^{2}}\right)^{\frac{1}{2}}$$
Answer

1. $$9$$

3. $$4$$

5. $$6x^{2}y$$

7. $$a^{1/3}$$

Exercise $$\PageIndex{11}$$ Solving Radical Equations

Solve.

1. $$\sqrt{x}=5$$
2. $$\sqrt{2x−1}=3$$
3. $$\sqrt{x−8}+2=5$$
4. $$\sqrt{3x−5}−1=11$$
5. $$\sqrt{5x−3}=\sqrt{2x+15}$$
6. $$\sqrt{8x−15}=x$$
7. $$\sqrt{x+41}=x−1$$
8. $$\sqrt{7−3x}=x−3$$
9. $$2(x+1)=\sqrt{2(x+1)}$$
10. $$\sqrt{x(x+6)}=4$$
11. $$\sqrt[3]{x(3x+10)}=2$$
12. $$\sqrt[3]{2x^{2}−x}+4=5$$
13. $$\sqrt[3]{3(x+4)(x+1)}=\sqrt[3]{5x+37}$$
14. $$\sqrt[3]{3x^{2}−9x+24}=\sqrt[3]{(x+2)^{2}}$$
15. $$y^{1/2}−3=0$$
16. $$y^{1/3}+3=0$$
17. $$(x−5)^{1/2}−2=0$$
18. $$(2x−1)^{1/3}−5=0$$
Answer

1. $$25$$

3. $$17$$

5. $$6$$

7. $$8$$

9. $$−\frac{1}{2}, −1$$

11. $$\frac{2}{3}, −4$$

13. $$−5, \frac{5}{3}$$

15. $$9$$

17. $$9$$

## Sample Exam

In Exercises 12-16, assume all variables represent nonnegative numbers.

Exercise $$\PageIndex{12}$$

Simplify.

1. $$\sqrt{100}$$
2. $$\sqrt{-100}$$
3. $$-\sqrt{100}$$
Answer

1. a. 10 b. Not a real number c. -10

Exercise $$\PageIndex{13}$$

Simplify.

1. g
1. $$\sqrt[3]{27}$$
2. $$\sqrt[3]{-27}$$
3. $$-\sqrt[3]{27}$$
2. $$\sqrt{\frac{128}{25}}$$
3. $$\sqrt[3]{\frac{192}{125}}$$
4. $$5 \sqrt{12 x^{2} y^{3} z}$$
5. $$2 \sqrt[3]{50 x^{2} y^{3} z^{5}}$$
Answer

2. $$\frac{8 \sqrt{2}}{5}$$

4. $$10xy\sqrt{3yz}$$

Exercise $$\PageIndex{14}$$

Perform the operations.

1. $$5 \sqrt{24}-\sqrt{108}+\sqrt{96}-3 \sqrt{27}$$
2. $$3 \sqrt{8 x^{2} y}-\left(x \sqrt{200 y}-\sqrt{18 x^{2} y}\right)$$
3. $$2 \sqrt{a b} \cdot(3 \sqrt{2 a}-\sqrt{b})$$
4. $$(\sqrt{x}−\sqrt{2y})^{2}$$
Answer

1. $$14 \sqrt{6}-15 \sqrt{3}$$

3. $$6a\sqrt{2b}−2b\sqrt{a}$$

Exercise $$\PageIndex{15}$$

Rationalize the denominator.

1. $$\frac{10}{\sqrt{2 x}}$$
2. $$\sqrt[3]{\frac{1}{4 x y^{2}}}$$
3. $$\frac{1}{\sqrt{x}+5}$$
4. $$\frac{\sqrt{2}-\sqrt{3}}{\sqrt{2}+\sqrt{3}}$$
Answer

1. $$\frac{5 \sqrt{2x}}{x}$$

3. $$\frac{\sqrt{x}-5}{x-25}$$

Exercise $$\PageIndex{16}$$

Perform the operations and simplify. Leave answers in exponential form.

1. $$2^{\frac{2}{3}} \cdot 2^{\frac{1}{6}}$$
2. $$\frac{10^{\frac{4}{5}}}{10^{\frac{1}{3}}}$$
3. $$\left(121 a^{4} b^{2}\right)^{\frac{1}{2}}$$
4. $$\frac{\left(9 y^{\frac{1}{3}} x^{6}\right)^{\frac{1}{2}}}{y^{\frac{1}{6}}}$$
Answer

1. $$2^{5/6}$$

3. $$11a^{2}b$$

Exercise $$\PageIndex{17}$$

Solve.

1. $$\sqrt{x}-7=0$$
2. $$\sqrt{3x+5}=1$$
3. $$\sqrt{2x−1}+2=x$$
4. $$3\sqrt{1−10x}=x−4$$
5. $$\sqrt{(2x+1)(3x+2)}=\sqrt{3(2x+1)}$$
6. $$\sqrt[3]{x(2x−15)}=3$$
7. The period, T, of a pendulum in seconds is given the formula $$T=2π\sqrt{L/32}$$, where L represents the length in feet. Calculate the length of a pendulum if the period is $$1^{1/2}$$ seconds. Round off to the nearest tenth.
Answer

1. 49

3. 5

5. $$-\frac{1}{2}, \frac{1}{3}$$

7. 1.8 feet