Intermediate Algebra

# Review Exercises

Intermediate AlgebraReview Exercises

## Simplify Expressions with Roots

Simplify Expressions with Roots

In the following exercises, simplify.

481.

$225225$ $−16−16$

482.

$−169−169$ $−8−8$

483.

$8383$ $814814$ $24352435$

484.

$−5123−5123$ $−814−814$ $−15−15$

Estimate and Approximate Roots

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

485.

$6868$ $843843$

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

486.

$3737$ $843843$ $12541254$

Simplify Variable Expressions with Roots

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

487.

$a33a33$
$b77b77$

488.

$a14a14$
$w24w24$

489.

$m84m84$
$n205n205$

490.

$121m20121m20$
$−64a2−64a2$

491.

$216a63216a63$
$32b20532b205$

492.

$144x2y2144x2y2$
$169w8y10169w8y10$
$8a51b638a51b63$

Use the Product Property to Simplify Radical Expressions

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

493.

$125 125$

494.

$675 675$

495.

$62536253$ $12861286$

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

496.

$a23a23$
$b83b83$
$c138c138$

497.

$80s1580s15$
$96a7596a75$
$128b76128b76$

498.

$96r3s396r3s3$
$80x7y6380x7y63$
$80x8y9480x8y94$

499.

$−325−325$
$−18−18$

500.

$8+968+96$
$2+4022+402$

Use the Quotient Property to Simplify Radical Expressions

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

501.

$72987298$ $2481324813$ $69646964$

502.

$y4y8y4y8$ $u21u115u21u115$ $v30v126v30v126$

503.

$300 m 5 64 300 m 5 64$

504.

$28p7q228p7q2$
$81s8t3381s8t33$
$64p15q12464p15q124$

505.

$27p2q108p4q327p2q108p4q3$
$16c5d7250c2d2316c5d7250c2d23$
$2m9n7128m3n62m9n7128m3n6$

506.

$80q55q80q55q$
$−625353−625353$
$80m745m480m745m4$

## Simplify Rational Exponents

Simplify expressions with $a1na1n$

In the following exercises, write as a radical expression.

507.

$r12r12$ $s13s13$ $t14t14$

In the following exercises, write with a rational exponent.

508.

$21p21p$ $8q48q4$ $436r6436r6$

In the following exercises, simplify.

509.

$6251462514$
$2431524315$
$32153215$

510.

$(−1,000)13(−1,000)13$
$−1,00013−1,00013$
$(1,000)−13(1,000)−13$

511.

$(−32)15(−32)15$
$(243)−15(243)−15$
$−12513−12513$

Simplify Expressions with $amnamn$

In the following exercises, write with a rational exponent.

512.

$r74r74$
$(2pq5)3(2pq5)3$
$(12m7n)34(12m7n)34$

In the following exercises, simplify.

513.

$25322532$
$9−329−32$
$(−64)23(−64)23$

514.

$−6432−6432$
$−64−32−64−32$
$(−64)32(−64)32$

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

515.

$652·612652·612$
$(b15)35(b15)35$
$w27w97w27w97$

516.

$a34·a−14a−104a34·a−14a−104$
$(27​b23​c−52b−73c12)13(27​b23​c−52b−73c12)13$

In the following exercises, simplify.

517.

$72−3272−32$
$7p3+2p37p3+2p3$
$5x3−3x35x3−3x3$

518.

$11b−511b+311b11b−511b+311b$
$811cd4+511cd4−911cd4811cd4+511cd4−911cd4$

519.

$48+2748+27$
$543+1283543+1283$
$654−323204654−323204$

520.

$80c7−20c780c7−20c7$
$2162r104+432r1042162r104+432r104$

521.

$3 75 y 2 + 8 y 48 − 300 y 2 3 75 y 2 + 8 y 48 − 300 y 2$

In the following exercises, simplify.

522.

$(56)(−12)(56)(−12)$
$(−2184)(−94)(−2184)(−94)$

523.

$(32x3)(718x2)(32x3)(718x2)$
$(−620a23)(−216a33)(−620a23)(−216a33)$

Use Polynomial Multiplication to Multiply Radical Expressions

In the following exercises, multiply.

524.

$11(8+411)11(8+411)$
$33(93+183)33(93+183)$

525.

$(3−27)(5−47)(3−27)(5−47)$
$(x3−5)(x3−3)(x3−5)(x3−3)$

526.

$( 2 7 − 5 11 ) ( 4 7 + 9 11 ) ( 2 7 − 5 11 ) ( 4 7 + 9 11 )$

527.

$(4+11)2(4+11)2$
$(3−25)2(3−25)2$

528.

$( 7 + 10 ) ( 7 − 10 ) ( 7 + 10 ) ( 7 − 10 )$

529.

$( 3 x 3 + 2 ) ( 3 x 3 − 2 ) ( 3 x 3 + 2 ) ( 3 x 3 − 2 )$

Divide Square Roots

In the following exercises, simplify.

530.

$48754875$
$813243813243$

531.

$320mn−545m−7n3320mn−545m−7n3$
$16x4y−23−54x−2y4316x4y−23−54x−2y43$

Rationalize a One Term Denominator

In the following exercises, rationalize the denominator.

532.

$8383$ $740740$ $82y82y$

533.

$11131113$ $75437543$ $33x2333x23$

534.

$144144$ $93249324$ $69x3469x34$

Rationalize a Two Term Denominator

In the following exercises, simplify.

535.

$7 2 − 6 7 2 − 6$

536.

$5 n − 7 5 n − 7$

537.

$x + 8 x − 8 x + 8 x − 8$

In the following exercises, solve.

538.

$4 x − 3 = 7 4 x − 3 = 7$

539.

$5 x + 1 = −3 5 x + 1 = −3$

540.

$4 x − 1 3 = 3 4 x − 1 3 = 3$

541.

$u − 3 + 3 = u u − 3 + 3 = u$

542.

$4 x + 5 3 − 2 = −5 4 x + 5 3 − 2 = −5$

543.

$( 8 x + 5 ) 1 3 + 2 = −1 ( 8 x + 5 ) 1 3 + 2 = −1$

544.

$y + 4 − y + 2 = 0 y + 4 − y + 2 = 0$

545.

$2 8 r + 1 − 8 = 2 2 8 r + 1 − 8 = 2$

In the following exercises, solve.

546.

$10 + 2 c = 4 c + 16 10 + 2 c = 4 c + 16$

547.

$2 x 2 + 9 x − 18 3 = x 2 + 3 x − 2 3 2 x 2 + 9 x − 18 3 = x 2 + 3 x − 2 3$

548.

$r + 6 = r + 8 r + 6 = r + 8$

549.

$x + 1 − x − 2 = 1 x + 1 − x − 2 = 1$

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 $s=As=A$ 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 $s=24ds=24d$ to find the speed of the vehicle before the brakes were applied. Round your answer to the nearest tenth.

In the following exercises, evaluate each function.

552.

$g(x)=6x+1,g(x)=6x+1,$ find
$g(4)g(4)$
$g(8)g(8)$

553.

$G(x)=5x−1,G(x)=5x−1,$ find
$G(5)G(5)$
$G(2)G(2)$

554.

$h(x)=x2−43,h(x)=x2−43,$ find
$h(−2)h(−2)$
$h(6)h(6)$

555.

For the function
$g(x)=4−4x4,g(x)=4−4x4,$ find
$g(1)g(1)$
$g(−3)g(−3)$

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.

$g ( x ) = 2 − 3 x g ( x ) = 2 − 3 x$

557.

$F ( x ) = x + 3 x − 2 F ( x ) = x + 3 x − 2$

558.

$f ( x ) = 4 x 2 − 16 3 f ( x ) = 4 x 2 − 16 3$

559.

$F ( x ) = 10 − 7 x 4 F ( x ) = 10 − 7 x 4$

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

560.

$g ( x ) = x + 4 g ( x ) = x + 4$

561.

$g ( x ) = 2 x g ( x ) = 2 x$

562.

$f ( x ) = x − 1 3 f ( x ) = x − 1 3$

563.

$f ( x ) = x 3 + 3 f ( x ) = x 3 + 3$

## 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.

$−100−100$
$−13−13$
$−45−45$

In the following exercises, add or subtract.

565.

$−50 + −18 −50 + −18$

566.

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

567.

$( 6 + i ) − ( −2 − 4 i ) ( 6 + i ) − ( −2 − 4 i )$

568.

$( −7 − −50 ) − ( −32 − −18 ) ( −7 − −50 ) − ( −32 − −18 )$

Multiply Complex Numbers

In the following exercises, multiply.

569.

$( −2 − 5 i ) ( −4 + 3 i ) ( −2 − 5 i ) ( −4 + 3 i )$

570.

$−6 i ( −3 − 2 i ) −6 i ( −3 − 2 i )$

571.

$−4 · −16 −4 · −16$

572.

$( 5 − −12 ) ( −3 + −75 ) ( 5 − −12 ) ( −3 + −75 )$

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

573.

$( −2 − 3 i ) 2 ( −2 − 3 i ) 2$

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

574.

$( 9 − 2 i ) ( 9 + 2 i ) ( 9 − 2 i ) ( 9 + 2 i )$

Divide Complex Numbers

In the following exercises, divide.

575.

$2 + i 3 − 4 i 2 + i 3 − 4 i$

576.

$−4 3 − 2 i −4 3 − 2 i$

Simplify Powers of i

In the following exercises, simplify.

577.

$i 48 i 48$

578.

$i 255 i 255$

Order a print copy

As an Amazon Associate we earn from qualifying purchases.

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.

• If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
• If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution: