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.

$i 255 i 255$