Lynn Marecek

Review Exercises

Simplify Expressions with Roots

Simplify Expressions with Roots

In the following exercises, simplify.

481.

ⓐ $\sqrt{225}$ ⓑ $− \sqrt{16}$

482.

ⓐ $− \sqrt{169}$ ⓑ $\sqrt{−8}$

483.

ⓐ $\sqrt[3]{8}$ ⓑ $\sqrt[4]{81}$ ⓒ $\sqrt[5]{243}$

484.

ⓐ $\sqrt[3]{−512}$ ⓑ $\sqrt[4]{−81}$ ⓒ $\sqrt[5]{−1}$

Estimate and Approximate Roots

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

485.

ⓐ $\sqrt{68}$ ⓑ $\sqrt[3]{84}$

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

486.

ⓐ $\sqrt{37}$ ⓑ $\sqrt[3]{84}$ ⓒ $\sqrt[4]{125}$

Simplify Variable Expressions with Roots

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

487.

ⓐ $\sqrt[3]{a^{3}}$
ⓑ $\sqrt[7]{b^{7}}$

488.

ⓐ $\sqrt{a^{14}}$
ⓑ $\sqrt{w^{24}}$

489.

ⓐ $\sqrt[4]{m^{8}}$
ⓑ $\sqrt[5]{n^{20}}$

490.

ⓐ $\sqrt{121 m^{20}}$
ⓑ $− \sqrt{64 a^{2}}$

491.

ⓐ $\sqrt[3]{216 a^{6}}$
ⓑ $\sqrt[5]{32 b^{20}}$

492.

ⓐ $\sqrt{144 x^{2} y^{2}}$
ⓑ $\sqrt{169 w^{8} y^{10}}$
ⓒ $\sqrt[3]{8 a^{51} b^{6}}$

Simplify Radical Expressions

Use the Product Property to Simplify Radical Expressions

In the following exercises, use the Product Property to simplify radical expressions.

493.

$\sqrt{125}$

494.

$\sqrt{675}$

495.

ⓐ $\sqrt[3]{625}$ ⓑ $\sqrt[6]{128}$

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

496.

ⓐ $\sqrt{a^{23}}$
ⓑ $\sqrt[3]{b^{8}}$
ⓒ $\sqrt[8]{c^{13}}$

497.

ⓐ $\sqrt{80 s^{15}}$
ⓑ $\sqrt[5]{96 a^{7}}$
ⓒ $\sqrt[6]{128 b^{7}}$

498.

ⓐ $\sqrt{96 r^{3} s^{3}}$
ⓑ $\sqrt[3]{80 x^{7} y^{6}}$
ⓒ $\sqrt[4]{80 x^{8} y^{9}}$

499.

ⓐ $\sqrt[5]{−32}$
ⓑ $\sqrt[8]{−1}$

500.

ⓐ $8 + \sqrt{96}$
ⓑ $\frac{2 + \sqrt{40}}{2}$

Use the Quotient Property to Simplify Radical Expressions

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

501.

ⓐ $\sqrt{\frac{72}{98}}$ ⓑ $\sqrt[3]{\frac{24}{81}}$ ⓒ $\sqrt[4]{\frac{6}{96}}$

502.

ⓐ $\sqrt{\frac{y^{4}}{y^{8}}}$ ⓑ $\sqrt[5]{\frac{u^{21}}{u^{11}}}$ ⓒ $\sqrt[6]{\frac{v^{30}}{v^{12}}}$

503.

$\sqrt{\frac{300 m^{5}}{64}}$

504.

ⓐ $\sqrt{\frac{28 p^{7}}{q^{2}}}$
ⓑ $\sqrt[3]{\frac{81 s^{8}}{t^{3}}}$
ⓒ $\sqrt[4]{\frac{64 p^{15}}{q^{12}}}$

505.

ⓐ $\sqrt{\frac{27 p^{2} q}{108 p^{4} q^{3}}}$
ⓑ $\sqrt[3]{\frac{16 c^{5} d^{7}}{250 c^{2} d^{2}}}$
ⓒ $\sqrt[6]{\frac{2 m^{9} n^{7}}{128 m^{3} n}}$

506.

ⓐ $\frac{\sqrt{80 q^{5}}}{\sqrt{5 q}}$
ⓑ $\frac{\sqrt[3]{- 625}}{\sqrt[3]{5}}$
ⓒ $\frac{\sqrt[4]{80 m^{7}}}{\sqrt[4]{5 m}}$

Simplify Rational Exponents

Simplify expressions with $a^{\frac{1}{n}}$

In the following exercises, write as a radical expression.

507.

ⓐ $r^{\frac{1}{2}}$ ⓑ $s^{\frac{1}{3}}$ ⓒ $t^{\frac{1}{4}}$

In the following exercises, write with a rational exponent.

508.

ⓐ $\sqrt{21 p}$ ⓑ $\sqrt[4]{8 q}$ ⓒ $4 \sqrt[6]{36 r}$

In the following exercises, simplify.

509.

ⓐ $625^{\frac{1}{4}}$
ⓑ $243^{\frac{1}{5}}$
ⓒ $32^{\frac{1}{5}}$

510.

ⓐ ${(−1,000)}^{\frac{1}{3}}$
ⓑ $− {1,000}^{\frac{1}{3}}$
ⓒ ${(1,000)}^{- \frac{1}{3}}$

511.

ⓐ ${(−32)}^{\frac{1}{5}}$
ⓑ ${(243)}^{- \frac{1}{5}}$
ⓒ $− 125^{\frac{1}{3}}$

Simplify Expressions with $a^{\frac{m}{n}}$

In the following exercises, write with a rational exponent.

512.

ⓐ $\sqrt[4]{r^{7}}$
ⓑ ${(\sqrt[5]{2 p q})}^{3}$
ⓒ $\sqrt[4]{{(\frac{12 m}{7 n})}^{3}}$

In the following exercises, simplify.

513.

ⓐ $25^{\frac{3}{2}}$
ⓑ $9^{- \frac{3}{2}}$
ⓒ ${(−64)}^{\frac{2}{3}}$

514.

ⓐ $− 64^{\frac{3}{2}}$
ⓑ $− 64^{- \frac{3}{2}}$
ⓒ ${(−64)}^{\frac{3}{2}}$

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

515.

ⓐ $6^{\frac{5}{2}} \cdot 6^{\frac{1}{2}}$
ⓑ ${(b^{15})}^{\frac{3}{5}}$
ⓒ $\frac{w^{\frac{2}{7}}}{w^{\frac{9}{7}}}$

516.

ⓐ $\frac{a^{\frac{3}{4}} \cdot a^{- \frac{1}{4}}}{a^{- \frac{10}{4}}}$
ⓑ ${(\frac{27 b^{\frac{2}{3}} c^{- \frac{5}{2}}}{b^{- \frac{7}{3}} c^{\frac{1}{2}}})}^{\frac{1}{3}}$

Add, Subtract and Multiply Radical Expressions

Add and Subtract Radical Expressions

In the following exercises, simplify.

517.

ⓐ $7 \sqrt{2} - 3 \sqrt{2}$
ⓑ $7 \sqrt[3]{p} + 2 \sqrt[3]{p}$
ⓒ $5 \sqrt[3]{x} - 3 \sqrt[3]{x}$

518.

ⓐ $\sqrt{11 b} - 5 \sqrt{11 b} + 3 \sqrt{11 b}$
ⓑ $8 \sqrt[4]{11 c d} + 5 \sqrt[4]{11 c d} - 9 \sqrt[4]{11 c d}$

519.

ⓐ $\sqrt{48} + \sqrt{27}$
ⓑ $\sqrt[3]{54} + \sqrt[3]{128}$
ⓒ $6 \sqrt[4]{5} - \frac{3}{2} \sqrt[4]{320}$

520.

ⓐ $\sqrt{80 c^{7}} - \sqrt{20 c^{7}}$
ⓑ $2 \sqrt[4]{162 r^{10}} + 4 \sqrt[4]{32 r^{10}}$

521.

$3 \sqrt{75 y^{2}} + 8 y \sqrt{48} - \sqrt{300 y^{2}}$

Multiply Radical Expressions

In the following exercises, simplify.

522.

ⓐ $(5 \sqrt{6}) (− \sqrt{12})$
ⓑ $(−2 \sqrt[4]{18}) (− \sqrt[4]{9})$

523.

ⓐ $(3 \sqrt{2 x^{3}}) (7 \sqrt{18 x^{2}})$
ⓑ $(−6 \sqrt[3]{20 a^{2}}) (−2 \sqrt[3]{16 a^{3}})$

Use Polynomial Multiplication to Multiply Radical Expressions

In the following exercises, multiply.

524.

ⓐ $\sqrt{11} (8 + 4 \sqrt{11})$
ⓑ $\sqrt[3]{3} (\sqrt[3]{9} + \sqrt[3]{18})$

525.

ⓐ $(3 - 2 \sqrt{7}) (5 - 4 \sqrt{7})$
ⓑ $(\sqrt[3]{x} - 5) (\sqrt[3]{x} - 3)$

526.

$(2 \sqrt{7} - 5 \sqrt{11}) (4 \sqrt{7} + 9 \sqrt{11})$

527.

ⓐ ${(4 + \sqrt{11})}^{2}$
ⓑ ${(3 - 2 \sqrt{5})}^{2}$

528.

$(7 + \sqrt{10}) (7 - \sqrt{10})$

529.

$(\sqrt[3]{3 x} + 2) (\sqrt[3]{3 x} - 2)$

Divide Radical Expressions

Divide Square Roots

In the following exercises, simplify.

530.

ⓐ $\frac{\sqrt{48}}{\sqrt{75}}$
ⓑ $\frac{\sqrt[3]{81}}{\sqrt[3]{24}}$

531.

ⓐ $\frac{\sqrt{320 m n^{−5}}}{\sqrt{45 m^{−7} n^{3}}}$
ⓑ $\frac{\sqrt[3]{16 x^{4} y^{−2}}}{\sqrt[3]{−54 x^{−2} y^{4}}}$

Rationalize a One Term Denominator

In the following exercises, rationalize the denominator.

532.

ⓐ $\frac{8}{\sqrt{3}}$ ⓑ $\sqrt{\frac{7}{40}}$ ⓒ $\frac{8}{\sqrt{2 y}}$

533.

ⓐ $\frac{1}{\sqrt[3]{11}}$ ⓑ $\sqrt[3]{\frac{7}{54}}$ ⓒ $\frac{3}{\sqrt[3]{3 x^{2}}}$

534.

ⓐ $\frac{1}{\sqrt[4]{4}}$ ⓑ $\sqrt[4]{\frac{9}{32}}$ ⓒ $\frac{6}{\sqrt[4]{9 x^{3}}}$

Rationalize a Two Term Denominator

In the following exercises, simplify.

535.

$\frac{7}{2 - \sqrt{6}}$

536.

$\frac{\sqrt{5}}{\sqrt{n} - \sqrt{7}}$

537.

$\frac{\sqrt{x} + \sqrt{8}}{\sqrt{x} - \sqrt{8}}$

Solve Radical Equations

Solve Radical Equations

In the following exercises, solve.

538.

$\sqrt{4 x - 3} = 7$

539.

$\sqrt{5 x + 1} = −3$

540.

$\sqrt[3]{4 x - 1} = 3$

541.

$\sqrt{u - 3} + 3 = u$

542.

$\sqrt[3]{4 x + 5} - 2 = −5$

543.

${(8 x + 5)}^{\frac{1}{3}} + 2 = −1$

544.

$\sqrt{y + 4} - y + 2 = 0$

545.

$2 \sqrt{8 r + 1} - 8 = 2$

Solve Radical Equations with Two Radicals

In the following exercises, solve.

546.

$\sqrt{10 + 2 c} = \sqrt{4 c + 16}$

547.

$\sqrt[3]{2 x^{2} + 9 x - 18} = \sqrt[3]{x^{2} + 3 x - 2}$

548.

$\sqrt{r} + 6 = \sqrt{r + 8}$

549.

$\sqrt{x + 1} - \sqrt{x - 2} = 1$

Use Radicals in Applications

In the following exercises, solve. Round approximations to one decimal place.

550.

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 answer to the nearest tenth of a foot.

551.

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 $s = \sqrt{24 d}$ to find the speed of the vehicle before the brakes were applied. Round your answer to the nearest tenth.

Use Radicals in Functions

Evaluate a Radical Function

In the following exercises, evaluate each function.

552.

$g (x) = \sqrt{6 x + 1},$ find
ⓐ $g (4)$
ⓑ $g (8)$

553.

$G (x) = \sqrt{5 x - 1},$ find
ⓐ $G (5)$
ⓑ $G (2)$

554.

$h (x) = \sqrt[3]{x^{2} - 4},$ find
ⓐ $h (−2)$
ⓑ $h (6)$

555.

For the function
$g (x) = \sqrt[4]{4 - 4 x},$ find
ⓐ $g (1)$
ⓑ $g (−3)$

Find the Domain of a Radical Function

In the following exercises, find the domain of the function and write the domain in interval notation.

556.

$g (x) = \sqrt{2 - 3 x}$

557.

$F (x) = \sqrt{\frac{x + 3}{x - 2}}$

558.

$f (x) = \sqrt[3]{4 x^{2} - 16}$

559.

$F (x) = \sqrt[4]{10 - 7 x}$

Graph Radical Functions

In the following exercises, ⓐ find the domain of the function ⓑ graph the function ⓒ use the graph to determine the range.

560.

$g (x) = \sqrt{x + 4}$

561.

$g (x) = 2 \sqrt{x}$

562.

$f (x) = \sqrt[3]{x - 1}$

563.

$f (x) = \sqrt[3]{x} + 3$

Use the Complex Number System

Evaluate the Square Root of a Negative Number

In the following exercises, write each expression in terms of i and simplify if possible.

564.

Add or Subtract Complex Numbers

In the following exercises, add or subtract.

565.

$\sqrt{−50} + \sqrt{−18}$

566.

$(8 - i) + (6 + 3 i)$

567.

$(6 + i) - (−2 - 4 i)$

568.

$(−7 - \sqrt{−50}) - (−32 - \sqrt{−18})$

Multiply Complex Numbers

In the following exercises, multiply.

569.

$(−2 - 5 i) (−4 + 3 i)$

570.

$−6 i (−3 - 2 i)$

571.

$\sqrt{−4} \cdot \sqrt{−16}$

572.

$(5 - \sqrt{−12}) (−3 + \sqrt{−75})$

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

573.

${(−2 - 3 i)}^{2}$

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

574.

$(9 - 2 i) (9 + 2 i)$

Divide Complex Numbers

In the following exercises, divide.

575.

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

576.

$\frac{−4}{3 - 2 i}$

Simplify Powers of i

In the following exercises, simplify.

577.

$i^{48}$

578.

$i^{255}$