8.E: Review Exercises and Sample Exam

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. 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