Search

Text Color

Margin Size

Font Type

Enable Dyslexic Font

5.E: Radical Functions and Equations (Exercises)

Last updated

Oct 6, 2021
Save as PDF
- 5.7: Complex Numbers and Their Operations
- 6: Solving Equations and Inequalities

Anonymous
LibreTexts

$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$

$\newcommand{\id}{\mathrm{id}}$ $\newcommand{\Span}{\mathrm{span}}$

( \newcommand{\kernel}{\mathrm{null}\,}\) $\newcommand{\range}{\mathrm{range}\,}$

$\newcommand{\RealPart}{\mathrm{Re}}$ $\newcommand{\ImaginaryPart}{\mathrm{Im}}$

$\newcommand{\Argument}{\mathrm{Arg}}$ $\newcommand{\norm}[1]{\| #1 \|}$

$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$\newcommand{\Span}{\mathrm{span}}$

$\newcommand{\id}{\mathrm{id}}$

$\newcommand{\Span}{\mathrm{span}}$

$\newcommand{\kernel}{\mathrm{null}\,}$

$\newcommand{\range}{\mathrm{range}\,}$

$\newcommand{\RealPart}{\mathrm{Re}}$

$\newcommand{\ImaginaryPart}{\mathrm{Im}}$

$\newcommand{\Argument}{\mathrm{Arg}}$

$\newcommand{\norm}[1]{\| #1 \|}$

$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$\newcommand{\Span}{\mathrm{span}}$ $\newcommand{\AA}{\unicode[.8,0]{x212B}}$

$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$

$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$

$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vectorC}[1]{\textbf{#1}}$

$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$

$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$

$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$

$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$

$\newcommand{\avec}{\mathbf a}$

$\newcommand{\bvec}{\mathbf b}$

$\newcommand{\cvec}{\mathbf c}$

$\newcommand{\dvec}{\mathbf d}$

$\newcommand{\dtil}{\widetilde{\mathbf d}}$

$\newcommand{\evec}{\mathbf e}$

$\newcommand{\fvec}{\mathbf f}$

$\newcommand{\nvec}{\mathbf n}$

$\newcommand{\pvec}{\mathbf p}$

$\newcommand{\qvec}{\mathbf q}$

$\newcommand{\svec}{\mathbf s}$

$\newcommand{\tvec}{\mathbf t}$

$\newcommand{\uvec}{\mathbf u}$

$\newcommand{\vvec}{\mathbf v}$

$\newcommand{\wvec}{\mathbf w}$

$\newcommand{\xvec}{\mathbf x}$

$\newcommand{\yvec}{\mathbf y}$

$\newcommand{\zvec}{\mathbf z}$

$\newcommand{\rvec}{\mathbf r}$

$\newcommand{\mvec}{\mathbf m}$

$\newcommand{\zerovec}{\mathbf 0}$

$\newcommand{\onevec}{\mathbf 1}$

$\newcommand{\real}{\mathbb R}$

$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$

$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$

$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$

$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$

$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$

$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$

$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$

$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$

$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$

$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$

$\newcommand{\bcal}{\cal B}$

$\newcommand{\ccal}{\cal C}$

$\newcommand{\scal}{\cal S}$

$\newcommand{\wcal}{\cal W}$

$\newcommand{\ecal}{\cal E}$

$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$

$\newcommand{\gray}[1]{\color{gray}{#1}}$

$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$

$\newcommand{\rank}{\operatorname{rank}}$

$\newcommand{\row}{\text{Row}}$

$\newcommand{\col}{\text{Col}}$

$\renewcommand{\row}{\text{Row}}$

$\newcommand{\nul}{\text{Nul}}$

$\newcommand{\var}{\text{Var}}$

$\newcommand{\corr}{\text{corr}}$

$\newcommand{\len}[1]{\left|#1\right|}$

$\newcommand{\bbar}{\overline{\bvec}}$

$\newcommand{\bhat}{\widehat{\bvec}}$

$\newcommand{\bperp}{\bvec^\perp}$

$\newcommand{\xhat}{\widehat{\xvec}}$

$\newcommand{\vhat}{\widehat{\vvec}}$

$\newcommand{\uhat}{\widehat{\uvec}}$

$\newcommand{\what}{\widehat{\wvec}}$

$\newcommand{\Sighat}{\widehat{\Sigma}}$

$\newcommand{\lt}{<}$

$\newcommand{\gt}{>}$

$\newcommand{\amp}{&}$

$\definecolor{fillinmathshade}{gray}{0.9}$

Exercise $\PageIndex{1}$

Simplify.

$- \sqrt { 121 }$
$\sqrt { ( - 7 ) ^ { 2 } }$
$\sqrt { ( x y ) ^ { 2 } }$
$\sqrt { ( 6 x - 7 ) ^ { 2 } }$
$\sqrt [ 3 ] { 125 }$
$\sqrt [ 3 ] { - 27 }$
$\sqrt [ 3 ] { ( x y ) ^ { 3 } }$
$\sqrt [ 3 ] { ( 6 x + 1 ) ^ { 3 } }$
Given $f ( x ) = \sqrt { x + 10 }$ , find $f(-1)$ and $f(6)$ .
Given $g(x) = \sqrt [ 3 ] { x - 5 }$ , find $g(4)$ and $g(13)$ .
Determine the domain of the function defined by $g ( x ) = \sqrt { 5 x + 2 }$ .
Determine the domain of the function defined by $g ( x ) = \sqrt [ 3 ] { 3 x - 1 }$ .

Answer

1. $-11$

3. $|xy|$

5. $5$

7. $xy$

9. $f ( - 1 ) = 3 ; f ( 6 ) = 4$

11. $\left[ - \frac { 2 } { 5 } , \infty \right)$

Exercise $\PageIndex{2}$

Simplify.

$\sqrt [ 3 ] { 250 }$
$4 \sqrt [ 3 ] { 120 }$
$- 3 \sqrt [ 3 ] { 108 }$
$10 \sqrt [ 5 ] { \frac { 1 } { 32 } }$
$- 6 \sqrt [ 4 ] { \frac { 81 } { 16 } }$
$\sqrt [ 6 ] { 128 }$
$\sqrt [ 5 ] { - 192 }$
$- 3 \sqrt { 420 }$

Answer

1. $5 \sqrt [ 3 ] { 2 }$

3. $- 9 \sqrt [ 3 ] { 4 }$

5. $-9$

7. $- 2 \sqrt [ 5 ] { 6 }$

Exercise $\PageIndex{3}$

Simplify.

$\sqrt { 20 x ^ { 4 } y ^ { 3 } }$
$- 4 \sqrt { 54 x ^ { 6 } y ^ { 3 } }$
$\sqrt { x ^ { 2 } - 14 x + 49 }$
$\sqrt { ( x - 8 ) ^ { 4 } }$

Answer

1. $2 x ^ { 2 } | y | \sqrt { 5 y }$

3. $| x - 7 |$

Exercise $\PageIndex{4}$

Simplify. (Assume all variable expressions are nonzero.)

$\sqrt { 100 x ^ { 2 } y ^ { 4 } }$
$\sqrt { 36 a ^ { 6 } b ^ { 2 } }$
$\sqrt { \frac { 8 a ^ { 2 } } { b ^ { 4 } } }$
$\sqrt { \frac { 72 x ^ { 4 } y } { z ^ { 6 } } }$
$10 x \sqrt { 150 x ^ { 7 } y ^ { 4 } }$
$- 5 n ^ { 2 } \sqrt { 25 m ^ { 10 } n ^ { 6 } }$
$\sqrt [ 3 ] { 48 x ^ { 6 } y ^ { 3 } z ^ { 2 } }$
$\sqrt [ 3 ] { 270 a ^ { 10 } b ^ { 8 } c ^ { 3 } }$
$\sqrt [ 3 ] { \frac { a ^ { 3 } b ^ { 5 } } { 64 c ^ { 6 } } }$
$\sqrt [ 5 ] { \frac { a ^ { 26 } } { 32 b ^ { 5 } c ^ { 10 } } }$
The period $T$ in seconds of a pendulum is given by the formula $T = 2 \pi \sqrt { \frac { L } { 32 } }$ where $L$ represents the length in feet of the pendulum. Calculate the period of a pendulum that is $2 \frac{1}{2}$ feet long. Give the exact answer and the approximate answer to the nearest hundredth of a second.
The time in seconds an object is in free fall is given by the formula $t = \frac { \sqrt { s } } { 4 }$ where $s$ represents the distance in feet the object has fallen. How long does it take an object to fall $28$ feet? Give the exact answer and the approximate answer to the nearest tenth of a second.
Find the distance between $(−5, 6)$ and $(−3,−4)$ .
Find the distance between $\left( \frac { 2 } { 3 } , - \frac { 1 } { 2 } \right)$ and $\left( 1 , - \frac { 3 } { 4 } \right)$ .

Answer

1. $10 x y ^ { 2 }$

3. $\frac { 2 a \sqrt { 2 } } { b ^ { 2 } }$

5. $50 x ^ { 4 } y ^ { 2 } \sqrt { 6 x }$

7. $2 x ^ { 2 } y \sqrt [ 3 ] { 6 z ^ { 2 } }$

9. $\frac { a b \sqrt [ 3 ] { b ^ { 2 } } } { 4 c ^ { 2 } }$

11. $\frac { \pi \sqrt { 5 } } { 4 }$ seconds; $1.76$ seconds

13. $2 \sqrt { 26 }$ units

Exercise $\PageIndex{5}$

Determine whether or not the three points form a right triangle. Use the Pythagorean theorem to justify your answer.

$( - 4,5 ) , ( - 3 , - 1 ) , \text { and } ( 3,0 )$
$( - 1 , - 1 ) , ( 1,3 ) , \text { and } ( - 6,1 )$

Answer: 1. Right triangle

Exercise $\PageIndex{6}$

Simplify. Assume all radicands containing variables are nonnegative.

$7 \sqrt { 2 } + 5 \sqrt { 2 }$
$8 \sqrt { 15 } - 2 \sqrt { 15 }$
$14 \sqrt { 3 } + 5 \sqrt { 2 } - 5 \sqrt { 3 } - 6 \sqrt { 2 }$
$22 \sqrt { a b } - 5 a \sqrt { b } + 7 \sqrt { a b } - 2 a \sqrt { b }$
$7 \sqrt { x } - ( 3 \sqrt { x } + 2 \sqrt { y } )$
$( 8 y \sqrt { x } - 7 x \sqrt { y } ) - ( 5 x \sqrt { y } - 12 y \sqrt { x } )$
$( 3 \sqrt { 5 } + 2 \sqrt { 6 } ) + ( 8 \sqrt { 5 } - 3 \sqrt { 6 } )$
$( 4 \sqrt [ 3 ] { 3 } - \sqrt [ 3 ] { 12 } ) - ( 5 \sqrt [ 3 ] { 3 } - 2 \sqrt [ 3 ] { 12 } )$
$( 2 - \sqrt { 10 x } + 3 \sqrt { y } ) - ( 1 + 2 \sqrt { 10 x } - 6 \sqrt { y } )$
$\left( 3 a \sqrt [ 3 ] { a b ^ { 2 } } + 6 \sqrt [ 3 ] { a ^ { 2 } b } \right) + \left( 9 a \sqrt [ 3 ] { a b ^ { 2 } } - 12 \sqrt [ 3 ] { a ^ { 2 } b } \right)$
$\sqrt { 45 } + \sqrt { 12 } - \sqrt { 20 } - \sqrt { 75 }$
$\sqrt { 24 } - \sqrt { 32 } + \sqrt { 54 } - 2 \sqrt { 32 }$
$2 \sqrt { 3 x ^ { 2 } } + \sqrt { 45 x } - x \sqrt { 27 } + \sqrt { 20 x }$
$5 \sqrt { 6 a ^ { 2 } b } + \sqrt { 8 a ^ { 2 } b ^ { 2 } } - 2 \sqrt { 24 a ^ { 2 } b } - a \sqrt { 18 b ^ { 2 } }$
$5 y \sqrt { 4 x ^ { 2 } y } - \left( x \sqrt { 16 y ^ { 3 } } - 2 \sqrt { 9 x ^ { 2 } y ^ { 3 } } \right)$
$\left( 2 b \sqrt { 9 a ^ { 2 } c } - 3 a \sqrt { 16 b ^ { 2 } c } \right) - \left( \sqrt { 64 a ^ { 2 } b ^ { 2 } c } - 9 b \sqrt { a ^ { 2 } c } \right)$
$\sqrt [ 3 ] { 216 x } - \sqrt [ 3 ] { 125 x y } - \sqrt [ 3 ] { 8 x }$
$\sqrt [ 3 ] { 128 x ^ { 3 } } - 2 x \sqrt [ 3 ] { 54 } + 3 \sqrt [ 3 ] { 2 x ^ { 3 } }$
$\sqrt [ 3 ] { 8 x ^ { 3 } y } - 2 x \sqrt [ 3 ] { 8 y } + \sqrt [ 3 ] { 27 x ^ { 3 } y } + x \sqrt [ 3 ] { y }$
$\sqrt [ 3 ] { 27 a ^ { 3 } b } - 3 \sqrt [ 3 ] { 8 a b ^ { 3 } } + a \sqrt [ 3 ] { 64 b } - b \sqrt [ 3 ] { a }$
Calculate the perimeter of the triangle formed by the following set of vertices: $\{ ( - 3 , - 2 ) , ( - 1,1 ) , ( 1 , - 2 ) \}$ .
Calculate the perimeter of the triangle formed by the following set of vertices: $\{ ( 0 , - 4 ) , ( 2,0 ) , ( - 3,0 ) \}$ .

Answer

1. $12 \sqrt { 2 }$

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

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

7. $11 \sqrt { 5 } - \sqrt { 6 }$

9. $1 - 3 \sqrt { 10 x } + 9 \sqrt { y }$

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

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

15. $12 x y \sqrt { y }$

17. $4 \sqrt [ 3 ] { x } - 5 \sqrt [ 3 ] { x y }$

19. $2 x \sqrt [ 3 ] { y }$

21. $4 + 2 \sqrt { 13 }$ units

Exercise $\PageIndex{7}$

Multiply.

$\sqrt { 6 } \cdot \sqrt { 15 }$
$( 4 \sqrt { 2 } ) ^ { 2 }$
$\sqrt { 2 } ( \sqrt { 2 } - \sqrt { 10 } )$
$( \sqrt { 5 } - \sqrt { 6 } ) ^ { 2 }$
$( 5 - \sqrt { 3 } ) ( 5 + \sqrt { 3 } )$
$( 2 \sqrt { 6 } + \sqrt { 3 } ) ( \sqrt { 2 } - 5 \sqrt { 3 } )$
$( \sqrt { a } - 5 \sqrt { b } ) ^ { 2 }$
$3 \sqrt { x y } ( \sqrt { x } - 2 \sqrt { y } )$
$\sqrt [ 3 ] { 3 a ^ { 2 } } \cdot \sqrt [ 3 ] { 18 a }$
$\sqrt [ 3 ] { 49 a ^ { 2 } b } \cdot \sqrt [ 3 ] { 7 a ^ { 2 } b ^ { 2 } }$

Answer

1. $3 \sqrt { 10 }$

3. $2 - 2 \sqrt { 5 }$

5. $22$

7. $a - 10 \sqrt { a b } + 25 b$

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

Exercise $\PageIndex{8}$

Divide. Assume all variables represent nonzero numbers and rationalize the denominator where appropriate.

$\frac { \sqrt { 72 } } { \sqrt { 9 } }$
$\frac { 10 \sqrt { 48 } } { \sqrt { 64 } }$
$\frac { 5 } { \sqrt { 5 } }$
$\frac { \sqrt { 15 } } { \sqrt { 2 } }$
$\frac { 3 } { 2 \sqrt { 6 } }$
$\frac { 2 + \sqrt { 5 } } { \sqrt { 10 } }$
$\frac { 18 } { \sqrt { 3 x } }$
$\frac { 2 \sqrt { 3 x } } { \sqrt { 6 x y } }$
$\frac { 1 } { \sqrt [ 3 ] { 3 x ^ { 2 } } }$
$\frac { 5 a b ^ { 2 } } { \sqrt [ 3 ] { 5 a ^ { 2 } b } }$
$\sqrt [ 3 ] { \frac { 5 x z ^ { 2 } } { 49 x ^ { 2 } y ^ { 2 } z } }$
$\frac { 1 } { \sqrt [ 5 ] { 8 x ^ { 4 } y ^ { 2 } z } }$
$\frac { 9 x ^ { 2 } y } { \sqrt [ 5 ] { 81 x y ^ { 2 } z ^ { 3 } } }$
$\sqrt [ 5 ] { \frac { 27 a b ^ { 3 } } { 15 a ^ { 4 } b c ^ { 2 } } }$
$\frac { 1 } { \sqrt { 5 } - \sqrt { 3 } }$
$\frac { \sqrt { 3 } } { \sqrt { 2 } + 1 }$
$\frac { - 3 \sqrt { 6 } } { 2 - \sqrt { 10 } }$
$\frac { \sqrt { x y } } { \sqrt { x } - \sqrt { y } }$
$\frac { \sqrt { 2 } - \sqrt { 6 } } { \sqrt { 2 } + \sqrt { 6 } }$
$\frac { \sqrt { a } + \sqrt { b } } { \sqrt { a } - \sqrt { b } }$
The base of a triangle measures $2 \sqrt{6}$ units and the height measures $3 \sqrt{15}$ units. Find the area of the triangle.
If each side of a square measures $5+2 \sqrt{10}$ units, find the area of the square.

Answer

1. $2 \sqrt { 2 }$

3. $\sqrt { 5 }$

5. $\frac { \sqrt { 6 } } { 4 }$

7. $\frac { 6 \sqrt { 3 x } } { x }$

9. $\frac { \sqrt [ 3 ] { 9 x } } { 3 x }$

11. $\frac { \sqrt [ 3 ] { 35 x ^ { 2 } y z } } { 7 x y }$

13. $\frac { 3 x y \sqrt [ 5 ] { 3 x ^ { 4 } y ^ { 3 } z ^ { 2 } } } { z }$

15. $\frac { \sqrt { 5 } + \sqrt { 3 } } { 2 }$

17. $\sqrt { 6 } + \sqrt { 15 }$

19. $- 2 + \sqrt { 3 }$

21. $9 \sqrt { 10 }$ square units

Exercise $\PageIndex{9}$

Express in radical form.

$11 ^ { 1 / 2 }$
$2 ^ { 2 / 3 }$
$x ^ { 3 / 5 }$
$a ^ { - 4 / 5 }$

Answer

1. $\sqrt { 11 }$

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

Exercise $\PageIndex{10}$

Write as a radical and then simplify.

$16 ^ { 1 / 2 }$
$72 ^ { 1 / 2 }$
$8 ^ { 2 / 3 }$
$32 ^ { 1 / 3 }$
$\left( \frac { 1 } { 9 } \right) ^ { 3 / 2 }$
$\left( \frac { 1 } { 216 } \right) ^ { - 1 / 3 }$

Answer

1. $4$

3. $4$

5. $\frac{1}{27}$

Exercise $\PageIndex{11}$

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

$6 ^ { 1 / 2 } \cdot 6 ^ { 3 / 2 }$
$3 ^ { 1 / 3 } \cdot 3 ^ { 1 / 2 }$
$\frac { 6 ^ { 5 / 2 } } { 6 ^ { 3 / 2 } }$
$\frac { 4 ^ { 3 / 4 } } { 4 ^ { 1 / 4 } }$
$\left( 64 x ^ { 6 } y ^ { 2 } \right) ^ { 1 / 2 }$
$\left( 27 x ^ { 12 } y ^ { 6 } \right) ^ { 1 / 3 }$
$\left( \frac { a ^ { 4 / 3 } } { a ^ { 1 / 2 } } \right) ^ { 2 / 5 }$
$\left( \frac { 16 x ^ { 4 / 3 } } { y ^ { 2 } } \right) ^ { 1 / 2 }$
$\frac { 56 x ^ { 3 / 4 } y ^ { 3 / 2 } } { 14 x ^ { 1 / 2 } y ^ { 2 / 3 } }$
$\frac { \left( 4 a ^ { 4 } b ^ { 2 / 3 } c ^ { 4 / 3 } \right) ^ { 1 / 2 } } { 2 a ^ { 2 } b ^ { 1 / 6 } c ^ { 2 / 3 } }$
$\left( 9 x ^ { - 4 / 3 } y ^ { 1 / 3 } \right) ^ { - 3 / 2 }$
$\left( 16 x ^ { - 4 / 5 } y ^ { 1 / 2 } z ^ { - 2 / 3 } \right) ^ { - 3 / 4 }$

Answer

1. $36$

3. $6$

5. $8 x ^ { 3 } y$

7. $a ^ { 1 / 3 }$

9. $4 x ^ { 1 / 4 } y ^ { 5 / 6 }$

11. $\frac { x ^ { 2 } } { 27 y ^ { 1 / 2 } }$

Exercise $\PageIndex{12}$

Perform the operations with mixed indices.

$\sqrt { y } \cdot \sqrt [ 5 ] { y ^ { 2 } }$
$\sqrt [ 3 ] { y } \cdot \sqrt [ 5 ] { y ^ { 3 } }$
$\frac { \sqrt [ 3 ] { y ^ { 2 } } } { \sqrt [ 3 ] { y } }$
$\sqrt { \sqrt [ 3 ] { y ^ { 2 } } }$

Answer

1. $\sqrt [ 10 ] { y ^ { 9 } }$

3. $\sqrt [ 15 ] { y ^ { 7 } }$

Exercise $\PageIndex{13}$

Solve.

$2 \sqrt { x } + 3 = 13$
$\sqrt { 3 x - 2 } = 4$
$\sqrt { x - 5 } + 4 = 8$
$5 \sqrt { x + 3 } + 7 = 2$
$\sqrt { 4 x - 3 } = \sqrt { 2 x + 15 }$
$\sqrt { 8 x - 15 } = x$
$x - 1 = \sqrt { 13 - x }$
$\sqrt { 4 x - 3 } = 2 x - 3$
$\sqrt { x + 5 } = 5 - \sqrt { x }$
$\sqrt { x + 3 } = 3 \sqrt { x } - 1$
$\sqrt { 2 ( x + 1 ) } - \sqrt { x + 2 } = 1$
$\sqrt { 6 - x } + \sqrt { x - 2 } = 2$
$\sqrt { 3 x - 2 } + \sqrt { x - 1 } = 1$
$\sqrt { 9 - x } = \sqrt { x + 16 } - 1$
$\sqrt [ 3 ] { 4 x - 3 } = 2$
$\sqrt [ 3 ] { x - 8 } = - 1$
$\sqrt [ 3 ] { x ( 3 x + 10 ) } = 2$
$\sqrt [ 3 ] { 2 x ^ { 2 } - x } + 4 = 5$
$\sqrt [ 3 ] { 3 ( x + 4 ) ( x + 1 ) } = \sqrt [ 3 ] { 5 x + 37 }$
$\sqrt [ 3 ] { 3 x ^ { 2 } - 9 x + 24 } = \sqrt [ 3 ] { ( x + 2 ) ^ { 2 } }$
$y ^ { 1 / 2 } - 3 = 0$
$y ^ { 1 / 3 } + 3 = 0$
$( x - 5 ) ^ { 1 / 2 } - 2 = 0$
$( 2 x - 1 ) ^ { 1 / 3 } - 5 = 0$
$( x - 1 ) ^ { 1 / 2 } = x ^ { 1 / 2 } - 1$
$( x - 2 ) ^ { 1 / 2 } - ( x - 6 ) ^ { 1 / 2 } = 2$
$( x + 4 ) ^ { 1 / 2 } - ( 3 x ) ^ { 1 / 2 } = - 2$
$( 5 x + 6 ) ^ { 1 / 2 } = 3 - ( x + 3 ) ^ { 1 / 2 }$
Solve for $g : t = \sqrt { \frac { 2 s } { g } }$ .
Solve for $x:y = \sqrt [ 3 ] { x + 4 } - 2$ ,
The period in seconds of a pendulum is given by the formula $T = 2 \pi \sqrt { \frac { L } { 32 } }$ where $L$ represents the length in feet of the pendulum. Find the length of a pendulum that has a period of $1 \frac{1}{2}$ seconds. Find the exact answer and the approximate answer rounded off to the nearest tenth of a foot.
The outer radius of a spherical shell is given by the formula $r = \sqrt [ 3 ] { \frac { 3 V } { 4 \pi } } + 2$ where $V$ represents the inner volume in cubic centimeters. If the outer radius measures $8$ centimeters, find the inner volume of the sphere.
The speed of a vehicle before the brakes are applied can be estimated by the length of the skid marks left on the road. On dry pavement, the speed $v$ in miles per hour can be estimated by the formula $v = 2 \sqrt { 6 d }$ , where $d$ represents the length of the skid marks in feet. Estimate the length of a skid mark if the vehicle is traveling $30$ miles per hour before the brakes are applied.
Find the real root of the function defined by $f ( x ) = \sqrt [ 3 ] { x - 3 } + 2$ .

Answer

1. $25$

3. $21$

5. $9$

7. $4$

9. $4$

11. $7$

13. $1$

15. $\frac{11}{4}$

17. $−4, \frac{2}{3}$

19. $−5, \frac{5}{3}$

21. $9$

23. $9$

25. $1$

27. $12$

29. $g = \frac { 2 s } { t ^ { 2 } }$

31. $\frac { 18 } { \pi ^ { 2 } }$ feet; $1.8$ feet

33. $37.5$ feet

Exercise $\PageIndex{14}$

Write the complex number in standard form $a+bi$ .

$5 - \sqrt { - 16 }$
$- \sqrt { - 25 } - 6$
$\frac { 3 + \sqrt { - 8 } } { 10 }$
$\frac { \sqrt { - 12 } - 4 } { 6 }$

Answer

1. $5 - 4 i$

3. $\frac { 3 } { 10 } + \frac { \sqrt { 2 } } { 5 } i$

Exercise $\PageIndex{15}$

Perform the operations.

$( 6 - 12 i ) + ( 4 + 7 i )$
$( - 3 + 2 i ) - ( 6 - 4 i )$
$\left( \frac { 1 } { 2 } - i \right) - \left( \frac { 3 } { 4 } - \frac { 3 } { 2 } i \right)$
$\left( \frac { 5 } { 8 } - \frac { 1 } { 5 } i \right) + \left( \frac { 3 } { 2 } - \frac { 2 } { 3 } i \right)$
$( 5 - 2 i ) - ( 6 - 7 i ) + ( 4 - 4 i )$
$( 10 - 3 i ) + ( 20 + 5 i ) - ( 30 - 15 i )$
$4 i ( 2 - 3 i )$
$( 2 + 3 i ) ( 5 - 2 i )$
$( 4 + i ) ^ { 2 }$
$( 8 - 3 i ) ^ { 2 }$
$( 3 + 2 i ) ( 3 - 2 i )$
$( - 1 + 5 i ) ( - 1 - 5 i )$
$\frac { 2 + 9 i } { 2 i }$
$\frac { i } { 1 - 2 i }$
$\frac { 4 + 5 i } { 2 - i }$
$\frac { 3 - 2 i } { 3 + 2 i }$
$10 - 5 ( 2 - 3 i ) ^ { 2 }$
$( 2 - 3 i ) ^ { 2 } - ( 2 - 3 i ) + 4$
$\left( \frac { 1 } { 1 - i } \right) ^ { 2 }$
$\left( \frac { 1 + 2 i } { 3 i } \right) ^ { 2 }$
$\sqrt { - 8 } ( \sqrt { 3 } - \sqrt { - 4 } )$
$( 1 - \sqrt { - 18 } ) ( 3 - \sqrt { - 2 } )$
$( \sqrt { - 5 } - \sqrt { - 10 } ) ^ { 2 }$
$( 1 - \sqrt { - 2 } ) ^ { 2 } - ( 1 + \sqrt { - 2 } ) ^ { 2 }$
Show that both $-5i$ and $5i$ satisfy $x^{2}+25=0$ .
Show that both $1-2i$ and $1+2i$ satisfy $x^{2}-2x+5=0$ .

Answer

1. $10 - 5 i$

3. $- \frac { 1 } { 4 } + \frac { 1 } { 2 } i$

5. $3+i$

7. $12+8i$

9. $15+8i$

11. $13$

13. $\frac{9}{2}-i$

15. $\frac { 3 } { 5 } + \frac { 14 } { 5 } i$

17. $35+60i$

19. $\frac{1}{2}i$

21. $4 \sqrt { 2 } + 2 i \sqrt { 6 }$

23. $- 15 + 10 \sqrt { 2 }$

25. Answer may vary

Sample Exam

Exercise $\PageIndex{16}$

Simplify. (Assume all variables are positive.)

$5 x \sqrt { 121 x ^ { 2 } y ^ { 4 } }$
$2 x y ^ { 2 } \sqrt [ 3 ] { - 64 x ^ { 6 } y ^ { 9 } }$
Calculate the distance between $(-5,-3)$ and $(-2,6)$ .
The time in seconds an object is in free fall is given by the formula $t = \frac { \sqrt { s } } { 4 }$ where $s$ represents the distance in feet that the object has fallen. If a stone is dropped into a $36$ -foot pit, how long will it take to hit the bottom of the pit?

Answer

1. $55 x ^ { 2 } y ^ { 2 }$

3. $3\sqrt{10}$ units

Exercise $\PageIndex{17}$

Perform the operations and simplify. (Assume all variables are positive and rationalize the denominator where appropriate.)

$\sqrt { 150 x y ^ { 2 } } - 2 \sqrt { 18 x ^ { 3 } } + y \sqrt { 24 x } + x \sqrt { 128 x }$
$3 \sqrt [ 3 ] { 16 x ^ { 3 } y ^ { 2 } } - \left( 2 x \sqrt [ 3 ] { 250 y ^ { 2 } } - \sqrt [ 3 ] { 54 x ^ { 3 } y ^ { 2 } } \right)$
$2 \sqrt { 2 } ( \sqrt { 2 } - 3 \sqrt { 6 } )$
$( \sqrt { 10 } - \sqrt { 5 } ) ^ { 2 }$
$\frac { \sqrt { 6 } } { \sqrt { 2 } + \sqrt { 3 } }$
$\frac { 2 x } { \sqrt { 2 x y } }$
$\frac { 1 } { \sqrt [ 5 ] { 8 x y ^ { 2 } z ^ { 4 } } }$
Simplify: $81 ^ { 3 / 4 }$ .
Express in radical form: $x ^ { - 3 / 5 }$ .

Answer

1. $7 y \sqrt { 6 x } + 2 x \sqrt { 2 x }$

3. $4 - 12 \sqrt { 3 }$

5. $- 2 \sqrt { 3 } + 3 \sqrt { 2 }$

7. $\frac { \sqrt [ 5 ] { 4 x ^ { 4 } y ^ { 3 } z } } { 2 x y z }$

9. $\frac { 1 } { \sqrt [ 5 ] { x ^ { 3 } } }$

Exercise $\PageIndex{18}$

Solve.

$\sqrt { x } - 5 = 1$
$\sqrt [ 3 ] { 5 x - 2 } + 6 = 4$
$5 \sqrt { 2 x + 5 } - 2 x = 11$
$\sqrt { 4 - 3 x } + 2 = x$
$\sqrt { 2 x + 5 } - \sqrt { x + 3 } = 2$
The time in seconds an object is in free fall is given by the formula $t = \frac { \sqrt { s } } { 4 }$ where $s$ represents the distance in feet that the object has fallen. If a stone is dropped into a pit and it takes $4$ seconds to reach the bottom, how deep is the pit?
The width in inches of a container is given by the formula $w = \frac { \sqrt [ 3 ] { 4 V } } { 2 } + 1$ where $V$ represents the inside volume in cubic inches of the container. What is the inside volume of the container if the width is $6$ inches?

Answer

2. $-\frac{6}{5}$

4. $\varnothing$

6. $256$ feet

Exercise $\PageIndex{19}$

Perform the operations and write the answer in standard form.

$\sqrt { - 3 } ( \sqrt { 6 } - \sqrt { - 3 } )$
$\frac { 4 + 3 i } { 2 - i }$
$6 - 3 ( 2 - 3 i ) ^ { 2 }$

Answer

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

3. $21 + 36 i$

Sample Exam

Support Center

How can we help?