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Mathematics LibreTexts

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.

      1. \(\sqrt{225}\)
      2. \(-\sqrt{16}\)
      1. \(-\sqrt{169}\)
      2. \(\sqrt{-8}\)
      1. \(\sqrt[3]{8}\)
      2. \(\sqrt[4]{81}\)
      3. \(\sqrt[5]{243}\)
      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 \(\PageIndex{2}\) Estimate and Approximate Roots

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

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

    1.

    1. \(8<\sqrt{68}<9\)
    2. \(4<\sqrt[3]{84}<5\)
    Exercise \(\PageIndex{3}\) Estimate and Approximate Roots

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

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

    1. Solve for yourself

    Exercise \(\PageIndex{4}\) Simplify Variable Expressions with Roots

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

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

    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.

    1. \(\sqrt{125}\)
    2. \(\sqrt{675}\)
      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 \(\PageIndex{6}\) Use the Product Property to Simplify Radical Expressions

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

      1. \(\sqrt{a^{23}}\)
      2. \(\sqrt[3]{b^{8}}\)
      3. \(\sqrt[8]{c^{13}}\)
      1. \(\sqrt{80 s^{15}}\)
      2. \(\sqrt[5]{96 a^{7}}\)
      3. \(\sqrt[6]{128 b^{7}}\)
      1. \(\sqrt{96 r^{3} s^{3}}\)
      2. \(\sqrt[3]{80 x^{7} y^{6}}\)
      3. \(\sqrt[4]{80 x^{8} y^{9}}\)
      1. \(\sqrt[5]{-32}\)
      2. \(\sqrt[8]{-1}\)
      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 \(\PageIndex{7}\) Use the Quotient Property to Simplify Radical Expressions

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

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

    Simplify Rational Exponents

    Exercise \(\PageIndex{8}\) Simplify Expressions with \(a^{\frac{1}{n}}\)

    In the following exercises, write as a radical expression.

      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 \(\PageIndex{9}\) Simplify Expressions with \(a^{\frac{1}{n}}\)

    In the following exercises, write with a rational exponent.

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

    1. Solve for yourself

    Exercise \(\PageIndex{10}\) Simplify Expressions with \(a^{\frac{1}{n}}\)

    In the following exercises, simplify.

      1. \(625^{\frac{1}{4}}\)
      2. \(243^{\frac{1}{5}}\)
      3. \(32^{\frac{1}{5}}\)
      1. \((-1,000)^{\frac{1}{3}}\)
      2. \(-1,000^{\frac{1}{3}}\)
      3. \((1,000)^{-\frac{1}{3}}\)
      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 \(\PageIndex{11}\) Simplify Expressions with \(a^{\frac{m}{n}}\)

    In the following exercises, write with a rational exponent.

      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 \(\PageIndex{12}\) Simplify Expressions with \(a^{\frac{m}{n}}\)

    In the following exercises, simplify.

      1. \(25^{\frac{3}{2}}\)
      2. \(9^{-\frac{3}{2}}\)
      3. \((-64)^{\frac{2}{3}}\)
      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 \(\PageIndex{13}\) Use the Laws of Exponents to Simplify Expressions with Rational Exponents

    In the following exercises, simplify.

      1. \(6^{\frac{5}{2}} \cdot 6^{\frac{1}{2}}\)
      2. \(\left(b^{15}\right)^{\frac{3}{5}}\)
      3. \(\frac{w^{\frac{2}{7}}}{w^{\frac{9}{7}}}\)
      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. \(\frac{1}{w}\)

    Add, Subtract and Multiply Radical Expressions

    Exercise \(\PageIndex{14}\) add and Subtract Radical Expressions

    In the following exercises, simplify.

      1. \(7 \sqrt{2}-3 \sqrt{2}\)
      2. \(7 \sqrt[3]{p}+2 \sqrt[3]{p}\)
      3. \(5 \sqrt[3]{x}-3 \sqrt[3]{x}\)
      1. \(\sqrt{11 b}-5 \sqrt{11 b}+3 \sqrt{11 b}\)
      2. \(8 \sqrt[4]{11 c d}+5 \sqrt[4]{11 c d}-9 \sqrt[4]{11 c d}\)
      1. \(\sqrt{48}+\sqrt{27}\)
      2. \(\sqrt[3]{54}+\sqrt[3]{128}\)
      3. \(6 \sqrt[4]{5}-\frac{3}{2} \sqrt[4]{320}\)
      1. \(\sqrt{80 c^{7}}-\sqrt{20 c^{7}}\)
      2. \(2 \sqrt[4]{162 r^{10}}+4 \sqrt[4]{32 r^{10}}\)
    1. \(3 \sqrt{75 y^{2}}+8 y \sqrt{48}-\sqrt{300 y^{2}}\)
    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 \(\PageIndex{15}\) Multiply Radical Expressions

    In the following exercises, simplify.

      1. \((5 \sqrt{6})(-\sqrt{12})\)
      2. \((-2 \sqrt[4]{18})(-\sqrt[4]{9})\)
      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 \(\PageIndex{16}\) Use Polynomial Multiplication to Multiply Radical Expressions

    In the following exercises, multiply.

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

    Divide Radical Expressions

    Exercise \(\PageIndex{17}\) Divide Square Roots

    In the following exercises, simplify.

      1. \(\frac{\sqrt{48}}{\sqrt{75}}\)
      2. \(\frac{\sqrt[3]{81}}{\sqrt[3]{24}}\)
      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 \(\PageIndex{18}\) rationalize a One Term Denominator

    In the following exercises, rationalize the denominator.

      1. \(\frac{8}{\sqrt{3}}\)
      2. \(\sqrt{\frac{7}{40}}\)
      3. \(\frac{8}{\sqrt{2 y}}\)
      1. \(\frac{1}{\sqrt[3]{11}}\)
      2. \(\sqrt[3]{\frac{7}{54}}\)
      3. \(\frac{3}{\sqrt[3]{3 x^{2}}}\)
      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 \(\PageIndex{19}\) Rationalize a Two Term Denominator

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

    Solve Radical Equations

    Exercise \(\PageIndex{20}\) Solve Radical Equations

    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 \(\PageIndex{21}\) Solve Radical Equations with Two Radicals

    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 \(\PageIndex{22}\) Use Radicals in Applications

    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

    Use Radicals in Functions

    Exercise \(\PageIndex{23}\) Evaluate a Radical Function

    In the following exercises, evaluate each function.

    1. \(g(x)=\sqrt{6 x+1}\), find
      1. \(g(4)\)
      2. \(g(8)\)
    2. \(G(x)=\sqrt{5 x-1}\), find
      1. \(G(5)\)
      2. \(G(2)\)
    3. \(h(x)=\sqrt[3]{x^{2}-4}\), find
      1. \(h(-2)\)
      2. \(h(6)\)
    4. For the function \(g(x)=\sqrt[4]{4-4 x}\), find
      1. \(g(1)\)
      2. \(g(-3)\)
    Answer

    2.

    1. \(G(5)=2 \sqrt{6}\)
    2. \(G(2)=3\)

    4.

    1. \(g(1)=0\)
    2. \(g(-3)=2\)
    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.

    1. \(g(x)=\sqrt{2-3 x}\)
    2. \(F(x)=\sqrt{\frac{x+3}{x-2}}\)
    3. \(f(x)=\sqrt[3]{4 x^{2}-16}\)
    4. \(F(x)=\sqrt[4]{10-7 x}\)
    Answer

    2. \((2, \infty)\)

    4. \(\left[\frac{7}{10}, \infty\right)\)

    Exercise \(\PageIndex{25}\) graph Radical Functions

    In the following exercises,

    1. find the domain of the function
    2. graph the function
    3. use the graph to determine the range
    1. \(g(x)=\sqrt{x+4}\)
    2. \(g(x)=2 \sqrt{x}\)
    3. \(f(x)=\sqrt[3]{x-1}\)
    4. \(f(x)=\sqrt[3]{x}+3\)
    Answer

    2.

    1. domain: \([0, \infty)\)

    2. The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from 0 to 8. The y-axis runs from 0 to 8. The function has a starting point at (0, 0) and goes through the points (1, 2) and (4, 4).
      Figure 8.E.1
    3. range: \([0, \infty)\)

    4.

    1. domain: \((-\infty, \infty)\)

    2. The figure shows a cube root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis runs from negative 2 to 6. The function has a center point at (0, 3) and goes through the points (negative 1, 2) and (1, 4).
      Figure 8.E.2
    3. 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.

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

    Solve for yourself

    Exercise \(\PageIndex{27}\) Add or Subtract Complex Numbers

    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 \(\PageIndex{28}\) Multiply Complex Numbers

    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 \(\PageIndex{29}\) Multiply Complex Numbers

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

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

    1. \(-5-12 i\)

    Exercise \(\PageIndex{30}\) Multiply Complex Numbers

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

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

    Solve for yourself

    Exercise \(\PageIndex{31}\) divide Complex Numbers

    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 \(\PageIndex{32}\) Simplify Powers of \(i\)

    In the following exercises, simplify.

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

    1. \(1\)

    Practice Test

    Exercise \(\PageIndex{33}\)

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

    1. \(\sqrt[3]{125 x^{9}}\)
    2. \(\sqrt{169 x^{8} y^{6}}\)
    3. \(\sqrt[3]{72 x^{8} y^{4}}\)
    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.

      1. \(216^{-\frac{1}{4}}\)
      2. \(-49^{\frac{3}{2}}\)
    1. \(\sqrt{-45}\)
    2. \(\frac{x^{-\frac{1}{4}} \cdot x^{\frac{5}{4}}}{x^{-\frac{3}{4}}}\)
    3. \(\left(\frac{8 x^{\frac{2}{3}} y^{-\frac{5}{2}}}{x^{-\frac{7}{3}} y^{\frac{1}{2}}}\right)^{\frac{1}{3}}\)
    4. \(\sqrt{48 x^{5}}-\sqrt{75 x^{5}}\)
    5. \(\sqrt{27 x^{2}}-4 x \sqrt{12}+\sqrt{108 x^{2}}\)
    6. \(2 \sqrt{12 x^{5}} \cdot 3 \sqrt{6 x^{3}}\)
    7. \(\sqrt[3]{4}(\sqrt[3]{16}-\sqrt[3]{6})\)
    8. \((4-3 \sqrt{3})(5+2 \sqrt{3})\)
    9. \(\frac{\sqrt[3]{128}}{\sqrt[3]{54}}\)
    10. \(\frac{\sqrt{245 x y^{-4}}}{\sqrt{45 x^{4} y^{3}}}\)
    11. \(\frac{1}{\sqrt[3]{5}}\)
    12. \(\frac{3}{2+\sqrt{3}}\)
    13. \(\sqrt{-4} \cdot \sqrt{-9}\)
    14. \(-4 i(-2-3 i)\)
    15. \(\frac{4+i}{3-2 i}\)
    16. \(i^{172}\)
    Answer

    1.

    1. \(\frac{1}{4}\)
    2. \(-343\)

    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.

    1. \(\sqrt{2 x+5}+8=6\)
    2. \(\sqrt{x+5}+1=x\)
    3. \(\sqrt[3]{2 x^{2}-6 x-23}=\sqrt[3]{x^{2}-3 x+5}\)
    Answer

    2. \(x=4\)

    Exercise \(\PageIndex{36}\)

    In the following exercise,

    1. find the domain of the function
    2. graph the function
    3. use the graph to determine the range
    1. \(g(x)=\sqrt{x+2}\)
    Answer

    1.

    1. domain: \([-2, \infty)\)

    2. The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 2 to 6. The y-axis runs from 0 to 8. The function has a starting point at (negative 2, 0) and goes through the points (negative 1, 1) and (2, 2).
      Figure 8.E.3
    3. range: \([0, \infty)\)
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