10.10: Chapter 8 Review Exercises
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Chapter Review Exercises
Simplify Expressions with Roots
Exercise \(\PageIndex{1}\) Simplify Expressions with Roots
In the following exercises, simplify.
-
- \(\sqrt{225}\)
- \(-\sqrt{16}\)
-
- \(-\sqrt{169}\)
- \(\sqrt{-8}\)
-
- \(\sqrt[3]{8}\)
- \(\sqrt[4]{81}\)
- \(\sqrt[5]{243}\)
-
- \(\sqrt[3]{-512}\)
- \(\sqrt[4]{-81}\)
- \(\sqrt[5]{-1}\)
- Answer
-
1.
- \(15\)
- \(-4\)
3.
- \(2\)
- \(3\)
- \(3\)
Exercise \(\PageIndex{2}\) Estimate and Approximate Roots
In the following exercises, estimate each root between two consecutive whole numbers.
-
- \(\sqrt{68}\)
- \(\sqrt[3]{84}\)
- Answer
-
1.
- \(8<\sqrt{68}<9\)
- \(4<\sqrt[3]{84}<5\)
Exercise \(\PageIndex{3}\) Estimate and Approximate Roots
In the following exercises, approximate each root and round to two decimal places.
-
- \(\sqrt{37}\)
- \(\sqrt[3]{84}\)
- \(\sqrt[4]{125}\)
- Answer
-
1. Solve for yourself
Exercise \(\PageIndex{4}\) Simplify Variable Expressions with Roots
In the following exercises, simplify using absolute values as necessary.
-
- \(\sqrt[3]{a^{3}}\)
- \(\sqrt[7]{b^{7}}\)
-
- \(\sqrt{a^{14}}\)
- \(\sqrt{w^{24}}\)
-
- \(\sqrt[4]{m^{8}}\)
- \(\sqrt[5]{n^{20}}\)
-
- \(\sqrt{121 m^{20}}\)
- \(-\sqrt{64 a^{2}}\)
-
- \(\sqrt[3]{216 a^{6}}\)
- \(\sqrt[5]{32 b^{20}}\)
-
- \(\sqrt{144 x^{2} y^{2}}\)
- \(\sqrt{169 w^{8} y^{10}}\)
- \(\sqrt[3]{8 a^{51} b^{6}}\)
- Answer
-
1.
- \(a\)
- \(|b|\)
3.
- \(m^{2}\)
- \(n^{4}\)
5.
- \(6a^{2}\)
- \(2b^{4}\)
Simplify Radical Expressions
Exercise \(\PageIndex{5}\) Use the Product Property to Simplify Radical Expressions
In the following exercises, use the Product Property to simplify radical expressions.
- \(\sqrt{125}\)
- \(\sqrt{675}\)
-
- \(\sqrt[3]{625}\)
- \(\sqrt[6]{128}\)
- Answer
-
1. \(5\sqrt{5}\)
3.
- \(5 \sqrt[3]{5}\)
- \(2 \sqrt[6]{2}\)
Exercise \(\PageIndex{6}\) Use the Product Property to Simplify Radical Expressions
In the following exercises, simplify using absolute value signs as needed.
-
- \(\sqrt{a^{23}}\)
- \(\sqrt[3]{b^{8}}\)
- \(\sqrt[8]{c^{13}}\)
-
- \(\sqrt{80 s^{15}}\)
- \(\sqrt[5]{96 a^{7}}\)
- \(\sqrt[6]{128 b^{7}}\)
-
- \(\sqrt{96 r^{3} s^{3}}\)
- \(\sqrt[3]{80 x^{7} y^{6}}\)
- \(\sqrt[4]{80 x^{8} y^{9}}\)
-
- \(\sqrt[5]{-32}\)
- \(\sqrt[8]{-1}\)
-
- \(8+\sqrt{96}\)
- \(\frac{2+\sqrt{40}}{2}\)
- Answer
-
2.
- \(4\left|s^{7}\right| \sqrt{5 s}\)
- \(2 a \sqrt[5]{3 a^{2}}\)
- \(2|b| \sqrt[6]{2 b}\)
4.
- \(-2\)
- not real
Exercise \(\PageIndex{7}\) Use the Quotient Property to Simplify Radical Expressions
In the following exercises, use the Quotient Property to simplify square roots.
-
- \(\sqrt{\frac{72}{98}}\)
- \(\sqrt[3]{\frac{24}{81}}\)
- \(\sqrt[4]{\frac{6}{96}}\)
-
- \(\sqrt{\frac{y^{4}}{y^{8}}}\)
- \(\sqrt[5]{\frac{u^{21}}{u^{11}}}\)
- \(\sqrt[6]{\frac{v^{30}}{v^{12}}}\)
- \(\sqrt{\frac{300 m^{5}}{64}}\)
-
- \(\sqrt{\frac{28 p^{7}}{q^{2}}}\)
- \(\sqrt[3]{\frac{81 s^{8}}{t^{3}}}\)
- \(\sqrt[4]{\frac{64 p^{15}}{q^{12}}}\)
-
- \(\sqrt{\frac{27 p^{2} q}{108 p^{4} q^{3}}}\)
- \(\sqrt[3]{\frac{16 c^{5} d^{7}}{250 c^{2} d^{2}}}\)
- \(\sqrt[6]{\frac{2 m^{9} n^{7}}{128 m^{3} n}}\)
-
- \(\frac{\sqrt{80 q^{5}}}{\sqrt{5 q}}\)
- \(\frac{\sqrt[3]{-625}}{\sqrt[3]{5}}\)
- \(\frac{\sqrt[4]{80 m^{7}}}{\sqrt[4]{5 m}}\)
- Answer
-
1.
- \(\frac{6}{7}\)
- \(\frac{2}{3}\)
- \(\frac{1}{2}\)
3. \(\frac{10 m^{2} \sqrt{3 m}}{8}\)
5.
- \(\frac{1}{2|p q|}\)
- \(\frac{2 c d \sqrt[5]{2 d^{2}}}{5}\)
- \(\frac{|m n| \sqrt[6]{2}}{2}\)
Simplify Rational Exponents
Exercise \(\PageIndex{8}\) Simplify Expressions with \(a^{\frac{1}{n}}\)
In the following exercises, write as a radical expression.
-
- \(r^{\frac{1}{2}}\)
- \(s^{\frac{1}{3}}\)
- \(t^{\frac{1}{4}}\)
- Answer
-
1.
- \(\sqrt{r}\)
- \(\sqrt[3]{s}\)
- \(\sqrt[4]{t}\)
Exercise \(\PageIndex{9}\) Simplify Expressions with \(a^{\frac{1}{n}}\)
In the following exercises, write with a rational exponent.
-
- \(\sqrt{21p}\)
- \(\sqrt[4]{8q}\)
- \(4\sqrt[6]{36r}\)
- Answer
-
1. Solve for yourself
Exercise \(\PageIndex{10}\) Simplify Expressions with \(a^{\frac{1}{n}}\)
In the following exercises, simplify.
-
- \(625^{\frac{1}{4}}\)
- \(243^{\frac{1}{5}}\)
- \(32^{\frac{1}{5}}\)
-
- \((-1,000)^{\frac{1}{3}}\)
- \(-1,000^{\frac{1}{3}}\)
- \((1,000)^{-\frac{1}{3}}\)
-
- \((-32)^{\frac{1}{5}}\)
- \((243)^{-\frac{1}{5}}\)
- \(-125^{\frac{1}{3}}\)
- Answer
-
1.
- \(5\)
- \(3\)
- \(2\)
3.
- \(-2\)
- \(\frac{1}{3}\)
- \(-5\)
Exercise \(\PageIndex{11}\) Simplify Expressions with \(a^{\frac{m}{n}}\)
In the following exercises, write with a rational exponent.
-
- \(\sqrt[4]{r^{7}}\)
- \((\sqrt[5]{2 p q})^{3}\)
- \(\sqrt[4]{\left(\frac{12 m}{7 n}\right)^{3}}\)
- Answer
-
1. Solve for yourself
Exercise \(\PageIndex{12}\) Simplify Expressions with \(a^{\frac{m}{n}}\)
In the following exercises, simplify.
-
- \(25^{\frac{3}{2}}\)
- \(9^{-\frac{3}{2}}\)
- \((-64)^{\frac{2}{3}}\)
-
- \(-64^{\frac{3}{2}}\)
- \(-64^{-\frac{3}{2}}\)
- \((-64)^{\frac{3}{2}}\)
- Answer
-
1.
- \(125\)
- \(\frac{1}{27}\)
- \(16\)
Exercise \(\PageIndex{13}\) Use the Laws of Exponents to Simplify Expressions with Rational Exponents
In the following exercises, simplify.
-
- \(6^{\frac{5}{2}} \cdot 6^{\frac{1}{2}}\)
- \(\left(b^{15}\right)^{\frac{3}{5}}\)
- \(\frac{w^{\frac{2}{7}}}{w^{\frac{9}{7}}}\)
-
- \(\frac{a^{\frac{3}{4}} \cdot a^{-\frac{1}{4}}}{a^{-\frac{10}{4}}}\)
- \(\left(\frac{27 b^{\frac{2}{3}} c^{-\frac{5}{2}}}{b^{-\frac{7}{3}} c^{\frac{1}{2}}}\right)^{\frac{1}{3}}\)
- Answer
-
1.
- \(6^{3}\)
- \(b^{9}\)
- \(\frac{1}{w}\)
Add, Subtract and Multiply Radical Expressions
Exercise \(\PageIndex{14}\) add and Subtract Radical Expressions
In the following exercises, simplify.
-
- \(7 \sqrt{2}-3 \sqrt{2}\)
- \(7 \sqrt[3]{p}+2 \sqrt[3]{p}\)
- \(5 \sqrt[3]{x}-3 \sqrt[3]{x}\)
-
- \(\sqrt{11 b}-5 \sqrt{11 b}+3 \sqrt{11 b}\)
- \(8 \sqrt[4]{11 c d}+5 \sqrt[4]{11 c d}-9 \sqrt[4]{11 c d}\)
-
- \(\sqrt{48}+\sqrt{27}\)
- \(\sqrt[3]{54}+\sqrt[3]{128}\)
- \(6 \sqrt[4]{5}-\frac{3}{2} \sqrt[4]{320}\)
-
- \(\sqrt{80 c^{7}}-\sqrt{20 c^{7}}\)
- \(2 \sqrt[4]{162 r^{10}}+4 \sqrt[4]{32 r^{10}}\)
- \(3 \sqrt{75 y^{2}}+8 y \sqrt{48}-\sqrt{300 y^{2}}\)
- Answer
-
1.
- \(4\sqrt{2}\)
- \(9\sqrt[3]{p}\)
- \(2\sqrt[3]{x}\)
3.
- \(7\sqrt{3}\)
- \(7\sqrt[3]{2}\)
- \(3\sqrt[4]{5}\)
5. \(37 y \sqrt{3}\)
Exercise \(\PageIndex{15}\) Multiply Radical Expressions
In the following exercises, simplify.
-
- \((5 \sqrt{6})(-\sqrt{12})\)
- \((-2 \sqrt[4]{18})(-\sqrt[4]{9})\)
-
- \(\left(3 \sqrt{2 x^{3}}\right)\left(7 \sqrt{18 x^{2}}\right)\)
- \(\left(-6 \sqrt[3]{20 a^{2}}\right)\left(-2 \sqrt[3]{16 a^{3}}\right)\)
- Answer
-
2.
- \(126 x^{2} \sqrt{2}\)
- \(48 a \sqrt[3]{a^{2}}\)
Exercise \(\PageIndex{16}\) Use Polynomial Multiplication to Multiply Radical Expressions
In the following exercises, multiply.
-
- \(\sqrt{11}(8+4 \sqrt{11})\)
- \(\sqrt[3]{3}(\sqrt[3]{9}+\sqrt[3]{18})\)
-
- \((3-2 \sqrt{7})(5-4 \sqrt{7})\)
- \((\sqrt[3]{x}-5)(\sqrt[3]{x}-3)\)
- \((2 \sqrt{7}-5 \sqrt{11})(4 \sqrt{7}+9 \sqrt{11})\)
-
- \((4+\sqrt{11})^{2}\)
- \((3-2 \sqrt{5})^{2}\)
- \((7+\sqrt{10})(7-\sqrt{10})\)
- \((\sqrt[3]{3 x}+2)(\sqrt[3]{3 x}-2)\)
- Answer
-
2.
- \(71-22 \sqrt{7}\)
- \(\sqrt[3]{x^{2}}-8 \sqrt[3]{x}+15\)
4.
- \(27+8 \sqrt{11}\)
- \(29-12 \sqrt{5}\)
6. \(\sqrt[3]{9 x^{2}}-4\)
Divide Radical Expressions
Exercise \(\PageIndex{17}\) Divide Square Roots
In the following exercises, simplify.
-
- \(\frac{\sqrt{48}}{\sqrt{75}}\)
- \(\frac{\sqrt[3]{81}}{\sqrt[3]{24}}\)
-
- \(\frac{\sqrt{320 m n^{-5}}}{\sqrt{45 m^{-7} n^{3}}}\)
- \(\frac{\sqrt[3]{16 x^{4} y^{-2}}}{\sqrt[3]{-54 x^{-2} y^{4}}}\)
- Answer
-
2.
- \(\frac{8 m^{4}}{3 n^{4}}\)
- \(-\frac{x^{2}}{2 y^{2}}\)
Exercise \(\PageIndex{18}\) rationalize a One Term Denominator
In the following exercises, rationalize the denominator.
-
- \(\frac{8}{\sqrt{3}}\)
- \(\sqrt{\frac{7}{40}}\)
- \(\frac{8}{\sqrt{2 y}}\)
-
- \(\frac{1}{\sqrt[3]{11}}\)
- \(\sqrt[3]{\frac{7}{54}}\)
- \(\frac{3}{\sqrt[3]{3 x^{2}}}\)
-
- \(\frac{1}{\sqrt[4]{4}}\)
- \(\sqrt[4]{\frac{9}{32}}\)
- \(\frac{6}{\sqrt[4]{9 x^{3}}}\)
- Answer
-
2.
- \(\frac{\sqrt[3]{121}}{11}\)
- \(\frac{\sqrt[3]{28}}{6}\)
- \(\frac{\sqrt[3]{9 x}}{x}\)
Exercise \(\PageIndex{19}\) Rationalize a Two Term Denominator
In the following exercises, simplify.
- \(\frac{7}{2-\sqrt{6}}\)
- \(\frac{\sqrt{5}}{\sqrt{n}-\sqrt{7}}\)
- \(\frac{\sqrt{x}+\sqrt{8}}{\sqrt{x}-\sqrt{8}}\)
- Answer
-
1. \(-\frac{7(2+\sqrt{6})}{2}\)
3. \(\frac{(\sqrt{x}+2 \sqrt{2})^{2}}{x-8}\)
Solve Radical Equations
Exercise \(\PageIndex{20}\) Solve Radical Equations
In the following exercises, solve.
- \(\sqrt{4 x-3}=7\)
- \(\sqrt{5 x+1}=-3\)
- \(\sqrt[3]{4 x-1}=3\)
- \(\sqrt{u-3}+3=u\)
- \(\sqrt[3]{4 x+5}-2=-5\)
- \((8 x+5)^{\frac{1}{3}}+2=-1\)
- \(\sqrt{y+4}-y+2=0\)
- \(2 \sqrt{8 r+1}-8=2\)
- Answer
-
2. no solution
4. \(u=3, u=4\)
6. \(x=-4\)
8. \(r=3\)
Exercise \(\PageIndex{21}\) Solve Radical Equations with Two Radicals
In the following exercises, solve.
- \(\sqrt{10+2 c}=\sqrt{4 c+16}\)
- \(\sqrt[3]{2 x^{2}+9 x-18}=\sqrt[3]{x^{2}+3 x-2}\)
- \(\sqrt{r}+6=\sqrt{r+8}\)
- \(\sqrt{x+1}-\sqrt{x-2}=1\)
- Answer
-
2. \(x=-8, x=2\)
4. \(x=3\)
Exercise \(\PageIndex{22}\) Use Radicals in Applications
In the following exercises, solve. Round approximations to one decimal place.
- 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.
- 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
Use Radicals in Functions
Exercise \(\PageIndex{23}\) Evaluate a Radical Function
In the following exercises, evaluate each function.
-
\(g(x)=\sqrt{6 x+1}\), find
- \(g(4)\)
- \(g(8)\)
-
\(G(x)=\sqrt{5 x-1}\), find
- \(G(5)\)
- \(G(2)\)
-
\(h(x)=\sqrt[3]{x^{2}-4}\), find
- \(h(-2)\)
- \(h(6)\)
-
For the function \(g(x)=\sqrt[4]{4-4 x}\), find
- \(g(1)\)
- \(g(-3)\)
- Answer
-
2.
- \(G(5)=2 \sqrt{6}\)
- \(G(2)=3\)
4.
- \(g(1)=0\)
- \(g(-3)=2\)
Exercise \(\PageIndex{24}\) Find the Domain of a Radical Function
In the following exercises, find the domain of the function and write the domain in interval notation.
- \(g(x)=\sqrt{2-3 x}\)
- \(F(x)=\sqrt{\frac{x+3}{x-2}}\)
- \(f(x)=\sqrt[3]{4 x^{2}-16}\)
- \(F(x)=\sqrt[4]{10-7 x}\)
- Answer
-
2. \((2, \infty)\)
4. \(\left[\frac{7}{10}, \infty\right)\)
Exercise \(\PageIndex{25}\) graph Radical Functions
In the following exercises,
- find the domain of the function
- graph the function
- use the graph to determine the range
- \(g(x)=\sqrt{x+4}\)
- \(g(x)=2 \sqrt{x}\)
- \(f(x)=\sqrt[3]{x-1}\)
- \(f(x)=\sqrt[3]{x}+3\)
- Answer
-
2.
- domain: \([0, \infty)\)
-
Figure 8.E.1 - range: \([0, \infty)\)
4.
- domain: \((-\infty, \infty)\)
-
Figure 8.E.2 - range: \((-\infty, \infty)\)
Use the Complex Number System
Exercise \(\PageIndex{26}\) evaluate the Square Root of a Negative Number
In the following exercises, write each expression in terms of \(i\) and simplify if possible.
-
- \(\sqrt{-100}\)
- \(\sqrt{-13}\)
- \(\sqrt{-45}\)
- Answer
-
Solve for yourself
Exercise \(\PageIndex{27}\) Add or Subtract Complex Numbers
In the following exercises, add or subtract.
- \(\sqrt{-50}+\sqrt{-18}\)
- \((8-i)+(6+3 i)\)
- \((6+i)-(-2-4 i)\)
- \((-7-\sqrt{-50})-(-32-\sqrt{-18})\)
- Answer
-
1. \(8 \sqrt{2} i\)
3. \(8+5 i\)
Exercise \(\PageIndex{28}\) Multiply Complex Numbers
In the following exercises, multiply.
- \((-2-5 i)(-4+3 i)\)
- \(-6 i(-3-2 i)\)
- \(\sqrt{-4} \cdot \sqrt{-16}\)
- \((5-\sqrt{-12})(-3+\sqrt{-75})\)
- Answer
-
1. \(23+14 i\)
3. \(-6\)
Exercise \(\PageIndex{29}\) Multiply Complex Numbers
In the following exercises, multiply using the Product of Binomial Squares Pattern.
- \((-2-3 i)^{2}\)
- Answer
-
1. \(-5-12 i\)
Exercise \(\PageIndex{30}\) Multiply Complex Numbers
In the following exercises, multiply using the Product of Complex Conjugates Pattern.
- \((9-2 i)(9+2 i)\)
- Answer
-
Solve for yourself
Exercise \(\PageIndex{31}\) divide Complex Numbers
In the following exercises, divide.
- \(\frac{2+i}{3-4 i}\)
- \(\frac{-4}{3-2 i}\)
- Answer
-
1. \(\frac{2}{25}+\frac{11}{25} i\)
Exercise \(\PageIndex{32}\) Simplify Powers of \(i\)
In the following exercises, simplify.
- \(i^{48}\)
- \(i^{255}\)
- Answer
-
1. \(1\)
Practice Test
Exercise \(\PageIndex{33}\)
In the following exercises, simplify using absolute values as necessary.
- \(\sqrt[3]{125 x^{9}}\)
- \(\sqrt{169 x^{8} y^{6}}\)
- \(\sqrt[3]{72 x^{8} y^{4}}\)
- \(\sqrt{\frac{45 x^{3} y^{4}}{180 x^{5} y^{2}}}\)
- Answer
-
1. \(5x^{3}\)
3. \(2 x^{2} y \sqrt[3]{9 x^{2} y}\)
Exercise \(\PageIndex{34}\)
In the following exercises, simplify. Assume all variables are positive.
-
- \(216^{-\frac{1}{4}}\)
- \(-49^{\frac{3}{2}}\)
- \(\sqrt{-45}\)
- \(\frac{x^{-\frac{1}{4}} \cdot x^{\frac{5}{4}}}{x^{-\frac{3}{4}}}\)
- \(\left(\frac{8 x^{\frac{2}{3}} y^{-\frac{5}{2}}}{x^{-\frac{7}{3}} y^{\frac{1}{2}}}\right)^{\frac{1}{3}}\)
- \(\sqrt{48 x^{5}}-\sqrt{75 x^{5}}\)
- \(\sqrt{27 x^{2}}-4 x \sqrt{12}+\sqrt{108 x^{2}}\)
- \(2 \sqrt{12 x^{5}} \cdot 3 \sqrt{6 x^{3}}\)
- \(\sqrt[3]{4}(\sqrt[3]{16}-\sqrt[3]{6})\)
- \((4-3 \sqrt{3})(5+2 \sqrt{3})\)
- \(\frac{\sqrt[3]{128}}{\sqrt[3]{54}}\)
- \(\frac{\sqrt{245 x y^{-4}}}{\sqrt{45 x^{4} y^{3}}}\)
- \(\frac{1}{\sqrt[3]{5}}\)
- \(\frac{3}{2+\sqrt{3}}\)
- \(\sqrt{-4} \cdot \sqrt{-9}\)
- \(-4 i(-2-3 i)\)
- \(\frac{4+i}{3-2 i}\)
- \(i^{172}\)
- Answer
-
1.
- \(\frac{1}{4}\)
- \(-343\)
3. \(x^{\frac{7}{4}}\)
5. \(-x^{2} \sqrt{3 x}\)
7. \(36 x^{4} \sqrt{2}\)
9. \(2-7 \sqrt{3}\)
11. \(\frac{7 x^{5}}{3 y^{7}}\)
13. \(3(2-\sqrt{3})\)
15. \(-12+8i\)
17. \(-i\)
Exercise \(\PageIndex{35}\)
In the following exercises, solve.
- \(\sqrt{2 x+5}+8=6\)
- \(\sqrt{x+5}+1=x\)
- \(\sqrt[3]{2 x^{2}-6 x-23}=\sqrt[3]{x^{2}-3 x+5}\)
- Answer
-
2. \(x=4\)
Exercise \(\PageIndex{36}\)
In the following exercise,
- find the domain of the function
- graph the function
- use the graph to determine the range
- \(g(x)=\sqrt{x+2}\)
- Answer
-
1.
- domain: \([-2, \infty)\)
-
Figure 8.E.3 - range: \([0, \infty)\)