8.E: Review Exercises and Sample Exam

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Review Exercises

(Assume all variables represent nonnegative numbers.)

Exercise \(\PageIndex{1}\) Radicals

Simplify.

\(\sqrt{36}\)
\(\sqrt{425}\)
\(\sqrt{−16}\)
\(-\sqrt{9}\)
\(\sqrt[3]{125}\)
\(3\sqrt[3]{−8}\)
\(\sqrt[3]{\frac{1}{64}}\)
\(−5\sqrt[3]{−27}\)
\(\sqrt{40}\)
\(−3\sqrt{50}\)
\(\sqrt{\frac{98}{81}}\)
\(\sqrt{\frac{11}{21}}\)
\(5\sqrt[3]{192}\)
\(2\sqrt[3]{−54}\)

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.

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

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.

\((5, −8)\) and \((2, −10)\)
\((−7, −1)\) and \((−6, 1)\)
\((−10, −1)\) and \((0, −5)\)
\((5, −1)\) and \((−2, −2)\)

Answer

1. \(\sqrt{13}\)

3. \(2\sqrt{29}\)

Exercise \(\PageIndex{4}\) Adding and Subtracting Radical Expressions

Simplify.

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

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.

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

Answer

1. \(3\sqrt{2}\)

3. \(\sqrt{6}-2 \sqrt{3}\)

5. \(−4\)

7. \(2a\)

Exercise \(\PageIndex{6}\) Multiplying and Dividing Radical Expressions

Divide.

\(\frac{\sqrt{72}}{\sqrt{4}}\)
\(10 \frac{\sqrt{48}}{\sqrt{64}}\)
\(\frac{\sqrt{98 x^{4} y^{2}}}{\sqrt{36 x^{2}}}\)
\(\frac{\sqrt[3]{81 x^{6} y^{7}}}{\sqrt[3]{8 y^{3}}}\)

Answer

1. \(3\sqrt{2}\)

3. \(\frac{7xy \sqrt{2}}{6}\)

Exercise \(\PageIndex{7}\) Multiplying and Dividing Radical Expressions

Rationalize the denominator.

\(\frac{2}{\sqrt{7}}\)
\(\frac{\sqrt{6}}{\sqrt{3}}\)
\(\sqrt{\frac{14}{2 x}}\)
\(\sqrt{\frac{12}{15}}\)
\(\sqrt[3]{\frac{1}{2 x^{2}}}\)
\(\sqrt[3]{\frac{5 a^{2} b^{5}}{a b^{2}}}\)
\(\frac{1}{\sqrt{3}-\sqrt{2}}\)
\(\frac{\sqrt{2}-\sqrt{6}}{\sqrt{2}+\sqrt{6}}\)

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.

\(7^{1/2}\)
\(3^{2/3}\)
\(x^{4/5}\)
\(y^{−3/4}\)

Answer

1. \(\sqrt{7}\)

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

Exercise \(\PageIndex{9}\)Rational Exponents

Write as a radical and then simplify.

\(4^{1/2}\)
\(50^{1/2}\)
\(4^{2/3}\)
\(81^{1/3}\)
\((\frac{1}{4})^{3/2}\)
\((\frac{12}{16})^{−1/3}\)

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.

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

Answer

1. \(9\)

3. \(4\)

5. \(6x^{2}y\)

7. \(a^{1/3}\)

Exercise \(\PageIndex{11}\) Solving Radical Equations

Solve.

\(\sqrt{x}=5\)
\(\sqrt{2x−1}=3\)
\(\sqrt{x−8}+2=5\)
\(\sqrt{3x−5}−1=11\)
\(\sqrt{5x−3}=\sqrt{2x+15}\)
\(\sqrt{8x−15}=x\)
\(\sqrt{x+41}=x−1\)
\(\sqrt{7−3x}=x−3\)
\(2(x+1)=\sqrt{2(x+1)}\)
\(\sqrt{x(x+6)}=4\)
\(\sqrt[3]{x(3x+10)}=2\)
\(\sqrt[3]{2x^{2}−x}+4=5\)
\(\sqrt[3]{3(x+4)(x+1)}=\sqrt[3]{5x+37}\)
\(\sqrt[3]{3x^{2}−9x+24}=\sqrt[3]{(x+2)^{2}}\)
\(y^{1/2}−3=0\)
\(y^{1/3}+3=0\)
\((x−5)^{1/2}−2=0\)
\((2x−1)^{1/3}−5=0\)

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.

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

Answer

2. \(\frac{8 \sqrt{2}}{5}\)

4. \(10xy\sqrt{3yz}\)

Exercise \(\PageIndex{14}\)

Perform the operations.

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

Answer

1. \(14 \sqrt{6}-15 \sqrt{3}\)

3. \(6a\sqrt{2b}−2b\sqrt{a}\)

Exercise \(\PageIndex{15}\)

Rationalize the denominator.

\(\frac{10}{\sqrt{2 x}}\)
\(\sqrt[3]{\frac{1}{4 x y^{2}}}\)
\(\frac{1}{\sqrt{x}+5}\)
\(\frac{\sqrt{2}-\sqrt{3}}{\sqrt{2}+\sqrt{3}}\)

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.

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

Answer

1. \(2^{5/6}\)

3. \(11a^{2}b\)

Exercise \(\PageIndex{17}\)

Solve.

\(\sqrt{x}-7=0\)
\(\sqrt{3x+5}=1\)
\(\sqrt{2x−1}+2=x\)
\(3\sqrt{1−10x}=x−4\)
\(\sqrt{(2x+1)(3x+2)}=\sqrt{3(2x+1)}\)
\(\sqrt[3]{x(2x−15)}=3\)
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