Review Exercises

Review 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

Simplify Radical Expressions

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

Add, Subtract and Multiply Radical Expressions

Add and Subtract Radical Expressions

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
ⓒ 654−32804654−32804

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

Multiply Radical Expressions

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 Radical Expressions

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

Solve Radical Equations

Solve Radical Equations

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

Solve Radical Equations with Two Radicals

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

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=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.

Use Radicals in Functions

Evaluate a Radical Function

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

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 ) = 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

Add or Subtract Complex Numbers

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