Algebra 2 6 Rational Exponents and Radical Functions ...rwright/algebra2/homework/Chapter 06 R… · Algebra 2 6 Rational Exponents and Radical Functions Practice Problems Page 4

Algebra 2 6 Rational Exponents and Radical Functions Practice Problems

Page 1 of 10

6.1 Evaluate nth Roots and Use Rational Exponents 1. In the expression √17

3, the 3 is called the ______________.

Rewrite the expression using rational exponent notation.

2. √73

3. √𝑥25

Rewrite the expression using radical notation.

4. 31

2 5. 54

3 6. (3𝑥)2

7

Evaluate the expression without using a calculator.

7. √325

8. (−8)2

3

9. √−1253

10. 16−3

4

Evaluate the expression using a calculator. Round to two decimal places if appropriate.

11. 643

4

12. √253610

13. 625−1

4

14. (√−8433

)2

Describe and correct the error in solving the equation. 15. 𝑥4 = 256

𝑥 = √2564

𝑥 = 4

Solve the equation. Round the solution to two decimal places. 16. 5𝑥3 = 1080

17. (𝑥 − 5)4 = 256

18. 7𝑥4 = 56

19. (𝑥 + 10)5 = 70

Word problem 20. A weir is a dam that is built across a river to regulate the flow of water.

The flow rate Q (in cubic feet per second) can be calculated using the

formula 𝑄 = 3.367ℓℎ3

2 where ℓ is the length (in feet) of the bottom of

the spillway and h is the depth (in feet) of the water on the spillway.

Determine the flow rate of a weir with a spillway that is 20 feet long

and has a water depth of 5 feet.

Mixed Review

21. (5.1) Simplify 𝑦11

4𝑧3 ⋅8𝑧7

𝑦7

22. (4.8) Solve 𝑥2 + 6𝑥 = −15

23. (4.2) Find the minimum or maximum value of the function 𝑔(𝑥) = −4(𝑥 + 6)2 − 12

24. (2.3) Graph 𝑦 = 𝑥 − 6

25. (1.7) Solve |3𝑝 − 6| = 21


Page 2 of 10

6.2 Apply Properties of Rational Exponents Simplify the expression.

1. 53

2 ⋅ 51

2

2. 80

14

5−

14

3. 120

−25⋅120

25

7−

34

4. √20 ⋅ √5

5. √84

⋅ √84

6. √645

√25

7. √364

⋅ √94

√44

8. 5√644

⋅ 2√84

9. √93

√275

10. 3

5√53

−1

5√53

11. 1

8√74

+3

8√74

12. −6√27

+ 2√2567

13. 2√12504

− 8√324

14. 𝑥1

4 ⋅ 𝑥1

3

15. 𝑥

25𝑦

𝑥𝑦−

13

16. √12𝑥2𝑦6𝑧124

17. 3

4𝑦

3

2 −1

4𝑦

3

2

18. 𝑦 √32𝑥64+ √162𝑥2𝑦44

Find the simplified expression for the perimeter and area of the given figure.

19. Word problem

20. The optimum diameter d (in millimeters) of the pinhole in a pinhole camera can be modeled by

𝑑 = 1.9[(5.5 × 10−4)ℓ]12

where ℓ is the length of the camera box (in millimeters). Find the optimum pinhole diameter for a camera

box with a length of 10 centimeters.

Mixed Review

21. (6.1) Rewrite using radical notation 71

3

22. (6.1) Evaluate without using a calculator 253

2

23. (5.3) Simplify (𝑦 − 7)(𝑦 + 6)

24. (3.8) Use an inverse matrix to solve the linear system {2𝑥 − 7𝑦 = −6−𝑥 + 5𝑦 = 3

25. (2.2) Find the slope of the line passing through (8, 9) and (-4, 3) and tell whether it rises, falls, is horizontal,

or vertical.


Page 3 of 10

6.3 Perform Function Operations and Compositions

Let 𝒇(𝒙) = −𝟑𝒙𝟏

𝟑 + 𝟒𝒙𝟏

𝟐 and 𝒈(𝒙) = 𝟓𝒙𝟏

𝟑 + 𝟒𝒙𝟏

𝟐. Perform the indicated operation. 1. 𝑓(𝑥) + 𝑔(𝑥)

2. 𝑓(𝑥) + 𝑓(𝑥)

3. 𝑓(𝑥) − 𝑔(𝑥)

Let 𝒇(𝒙) = 𝟒𝒙𝟐

𝟑 and 𝒈(𝒙) = 𝟓𝒙𝟏

𝟐. Perform the indicated operation. 4. 𝑔(𝑥) ⋅ 𝑓(𝑥)

5. 𝑔(𝑥) ⋅ 𝑔(𝑥) 6.

𝑔(𝑥)

𝑓(𝑥)

7. 𝑔(𝑥)

𝑔(𝑥)

Let 𝒇(𝒙) = 𝟑𝒙 + 𝟐, 𝒈(𝒙) = −𝒙𝟐, and 𝒉(𝒙) =𝒙−𝟐

𝟓. Find the indicated value.

8. 𝑔(𝑓(2)) 9. 𝑔(ℎ(8))

Let 𝒇(𝒙) = 𝟑𝒙−𝟏, 𝒈(𝒙) = 𝟐𝒙 − 𝟕, and 𝒉(𝒙) =𝒙+𝟒

𝟑. Perform the indicated operation

10. 𝑔(𝑓(𝑥))

11. 𝑔(ℎ(𝑥))

12. 𝑓(𝑓(𝑥))

13. 𝑔(𝑔(𝑥))

Let 𝒇(𝒙) = 𝒙𝟐 − 𝟑 and 𝒈(𝒙) = 𝟒𝒙. Describe and correct the error in the composition.

14. 𝑔(𝑓(𝑥)) = 𝑔(𝑥2 − 3)

= 4𝑥2 − 3

Word problem 15. An online movie store is having a sale. You decide to open a charge account and buy four movies. There are

two sales and you can use both! The first sale is $15 off any four movies, and the second is 10% off your

entire purchase when you open a charge account.

a. Use composition of functions to find the sale price of $85 worth of movies when the $15 discount is

applied before the 10% discount.

b. Use composition of functions to find the sale price of $85 worth of movies when the 10% discount is

applied before the $15 discount.

c. Which order of discounts gives you a better deal? Explain.

Mixed Review

16. (6.2) Simplify √1

6

3

17. (6.2) Simplify √𝑥15

𝑦6

3

18. (6.1) Solve the equation 𝑥3 = 125

19. (5.4) Factor completely 16𝑥3 − 44𝑥2 − 42𝑥

20. (4.6) Simplify (6 − 3𝑖) + (5 + 4𝑖)


Page 4 of 10

6.4 Use Inverse Functions 1. State the definition of an inverse relation.

Find an equation for the inverse relation. 2. 𝑦 = 4𝑥 − 1

3. 𝑦 = 12𝑥 + 7 4. 𝑦 = −

3

5𝑥 +

7

5

Verify that f and g are inverse functions. 5. 𝑓(𝑥) = 𝑥 + 4, 𝑔(𝑥) = 𝑥 − 4 6. 𝑓(𝑥) = 4𝑥 + 9, 𝑔(𝑥) =

1

4𝑥 −

9

4

Find the inverse of the power function. 7. 𝑓(𝑥) = 4𝑥4, 𝑥 ≥ 0 8. 𝑓(𝑥) =

16

25𝑥2, 𝑥 ≤ 0

Graph the function f. Then use the graph to determine whether the inverse of f is a function.

9. 𝑓(𝑥) =1

4𝑥2 − 1 10. 𝑓(𝑥) = (𝑥 − 4)(𝑥 + 1)

Find the inverse of the function. 11. 𝑓(𝑥) = 𝑥3 − 2

12. 𝑓(𝑥) = −2

5𝑥6 + 8, 𝑥 ≤ 0

13. 𝑓(𝑥) = 𝑥4 − 9, 𝑥 ≥ 0

Word problems 14. Show that the inverse of any linear function 𝑓(𝑥) = 𝑚𝑥 + 𝑏, where 𝑚 ≠ 0, is also a linear function. Give the

slope and y-intercept of the graph of 𝑓−1 in terms of m and b.

15. The maximum hull speed v (in knots) of a boat with a

displacement hull can be approximated by

𝑣 = 1.34√ℓ

where ℓ is the length (in feet) of the boat’s waterline. Find the

inverse of the model. Then find the waterline length needed to

achieve a maximum speed of 7.5 knots.

Mixed Review

16. (6.3) If 𝑓(𝑥) = −3𝑥1

3 + 4𝑥1

2 and 𝑔(𝑥) = 5𝑥1

3 + 4𝑥1

2, what is 𝑔(𝑥) − 𝑓(𝑥)?

17. (6.3) What is 𝑔(𝑓(𝑥)) if 𝑓(𝑥) = 7𝑥2 and 𝑔(𝑥) = 3𝑥−2?

18. (6.2) Simplify 3√𝑥5

+ 9√𝑥5

19. (6.1) Rewrite using radical notation. 𝑥7

6

20. (4.1) Graph 𝑓(𝑥) =1

2𝑥2 + 𝑥 − 3


Page 5 of 10

6.5 Graph Square Root and Cube Root Functions Graph the function. Then state the domain and range.

1. 𝑦 = −4√𝑥

2. 𝑦 = −4

5√𝑥

3. 𝑦 = 5√𝑥

4. The graph of which function is shown?

A. 𝑦 =3

4√𝑥 B. 𝑦 = −

3

4√𝑥 C. 𝑦 =

3

2√𝑥 D. 𝑦 = −

3

2√𝑥

5. 𝑦 = 2√𝑥3

6. ℎ(𝑥) = −1

7√𝑥3

7. 𝑦 =7

9√𝑥3

8. 𝑦 = (𝑥 + 1)1

2 + 8

9. 𝑦 =3

4𝑥

1

3 − 1

10. ℎ(𝑥) = −3√𝑥 + 73

− 6

11. 𝑔(𝑥) = −1

3√𝑥3

− 6

Word problems

12. Explain why there are limitations on the domain and range of the function 𝑦 = √𝑥 − 5 + 4.

13. If the graph of 𝑦 = 3√𝑥3

is shifted left 2 units, what is the equation of the translated graph?

14. The speed v (in meters per second) of sound waves in air depends on the temperature K (in kelvins) and

can be modeled by:

𝑣 = 331.5√𝐾

273.15, 𝐾 ≥ 0

a. Kelvin temperature K is related to Celsius temperature C by the formula 𝐾 = 273.15 + 𝐶. Write an

equation that gives the speed v of sound waves in air as a function of the temperature C in degrees Celsius.

b. What are a reasonable domain and range for the function from part (a)?

Mixed Review 15. (6.4) Find the inverse function 𝑓(𝑥) = −2𝑥 + 5

16. (6.4) Find the inverse function 𝑓(𝑥) =3

2𝑥4, 𝑥 ≤ 0

17. (6.3) If 𝑓(𝑥) = 3𝑥−1 and 𝑔(𝑥) = 2𝑥 − 7, find 𝑓(𝑔(𝑥)).

18. (6.2) Simplify √49𝑥5.

19. (4.8) Solve using the quadratic formula 𝑥2 − 5𝑥 + 10 = 4.

20. (1.6) Solve 𝑥 + 4 > 10.


Page 6 of 10

6.6 Solve Radical Equations 1. Copy and complete: When you solve an equation algebraically, an apparent solution that must be rejected

because it does not satisfy the original equation is called a(n) __________ solution.

Solve the equation. Check your solution.

2. √9𝑥 + 11 = 14

3. √𝑥 − 25 + 3 = 5

4. √𝑥3

− 10 = −3

5. −5√8𝑥3

+ 12 = −8

6. −4√𝑥 + 103

+ 3 = 15

7. 9𝑥2

5 = 36

8. (1

3𝑥 − 11)

1

2= 5

Describe and correct the error in solving the equation.

9. (𝑥 + 7)1

2 = 5

[(𝑥 + 7)1

2]2

= 5

𝑥 + 7 = 5

𝑥 = −2

Solve the equation. Check your solution.

10. √21𝑥 + 1 = 𝑥 + 5

11. √3 − 8𝑥24= 2𝑥

12. √4𝑥 + 1 = √𝑥 + 10

13. √𝑥 + 2 = 2 − √𝑥

Solve the system of equations.

14. {3√𝑥 + 5√𝑦 = 31

5√𝑥 − 5√𝑦 = −15

Word problem 15. A burning candle has a radius of r inches and was initially h0 inches tall. After t minutes,

the height of the candle has been reduced to h inches. These quantities are related by the

formula

𝑟 = √𝑘𝑡

𝜋(ℎ0 − ℎ)

where k is a constant. How long will it take for the entire candle to burn if its radius is

0.875 inch, its initial height is 6.5 inches, and k = 0.04?

Mixed Review

16. (6.5) Graph 𝑦 =1

4√𝑥3

.

17. (6.5) Graph 𝑦 = 2√𝑥 − 1 + 3

18. (6.4) Verify that f and g are inverse functions. 𝑓(𝑥) =1

4𝑥3, 𝑔(𝑥) = (4𝑥)

1

3

19. (6.3) Let 𝑓(𝑥) = 4𝑥2

3 and 𝑔(𝑥) = 5𝑥1

2. Find 𝑓(𝑥) ⋅ 𝑔(𝑥).

20. (6.1) Evaluate (−125)1

3


Page 7 of 10

Chapter 6 Review 1. Evaluate √150

4 using a calculator. Round the result to two decimal places if appropriate.

2. The volume of a sphere is given by 𝑉 =4

3𝜋𝑟3, where V is the volume and r is the radius of the sphere. Find

the radius of a sphere with a volume 4 ft3.

Simplify the expression. Assume all variables are positive.

3. 𝑞7

3 ⋅ 𝑞2

3

4. 𝑥10

3𝑥6

5. √813

+ √243

6. √64𝑥8𝑦105

Let 𝒇(𝒙) = 𝒙 + 𝟐, and 𝒈(𝒙) = 𝒙𝟐. Perform the indicated operation. 7. 𝑓(𝑥) − 𝑔(𝑥)

8. 𝑓(𝑥) ⋅ 𝑔(𝑥)

9. 𝑔(𝑓(𝑥))

Find the inverse of the function. 10. 𝑓(𝑥) = 64𝑥3

11. 𝑔(𝑥) = 𝑥10 − 2, 𝑥 ≤ 0

12. ℎ(𝑥) = 2(𝑥)4, 𝑥 ≥ 0

Graph the function. Then state the domain and range.

13. 𝑦 = √𝑥

14. 𝑦 = √𝑥3

15. 𝑦 = −2√𝑥3

+ 1

16. 𝑦 = √𝑥 − 2 − 3

Solve the equation.

17. √𝑥 + 2 = 10

18. 2√3𝑥 − 43

= 6

19. (𝑥 + 3)2

3 − 3 = 1

20. √𝑥 + 10 = 𝑥 + 1

21. √2𝑥 + 1 − √𝑥 − 3 = 0


Page 8 of 10

Answers

6.1 1. index

2. 71

3

3. 𝑥2

5

4. √3

5. √543

6. √(3𝑥)27

7. 2 8. 4 9. −5

10. 1

8

11. 22.63

12. 2.19 13. 0.2 14. 89.24 15. There are two solutions ±4 16. 6 17. 1, 9 18. ±1.68 19. -7.66 20. about 753 ft3/s 21. 2𝑧4𝑦4

22. −3 ± √6𝑖 23. max: -12

24. 25. -5, 9

6.2 1. 25

2. 2 ⋅ 51

2

3. 73

4 4. 10

5. 2√2 6. 2 7. 3

8. 40√24

9. √315

10. 2

5√53

11. 1

2√74

12. −2√27

13. −6√24

14. 𝑥7

12

15. 𝑦

43

𝑥35

16. 𝑦𝑧3 √12𝑥2𝑦24

17. 1

2𝑦

3

2

18. (2𝑥𝑦 + 3𝑦)√2𝑥24

19. perimeter: 24𝑥1

4; area: 35𝑥1

2 20. 0.45 mm

21. √73

22. 125 23. 𝑦2 − 𝑦 − 42 24. (-3, 0)

25. 1

2; rises

6.3 1. 2𝑥

1

3 + 8𝑥1

2

2. −6𝑥1

3 + 8𝑥1

2

3. −8𝑥1

3

4. 20𝑥7

6 5. 25x

6. 5

4𝑥16

7. 1

8. -64

9. −36

25

10. 6

𝑥− 7

11. 2𝑥−13

3

12. x 13. 4𝑥 − 21 14. 4 should be distributed to all terms. 4𝑥2 − 12

15. $63; $61.50; Apply the 10% discount first since that means you pay less money.

16. √363

6

17. 𝑥5

𝑦2

18. 5 19. 2𝑥(2𝑥 − 7)(4𝑥 + 3) 20. 11 + 𝑖

6.4 1. An inverse relation interchanges the input and output values of the original relation.

2. 𝑦 =𝑥+1

4

3. 𝑦 =𝑥−7

12

4. 𝑦 =7−5𝑥

3

5. Show work 6. Show work

7. 𝑓−1(𝑥) = √𝑥

4

4

8. 𝑓−1(𝑥) = −5

4√𝑥

9. not a function 10. not a function

11. 𝑓−1(𝑥) = √𝑥 + 23

12. 𝑓−1(𝑥) = −√40−5𝑥

2

6

13. 𝑓−1(𝑥) = √𝑥 + 94

14. 𝑓−1(𝑥) =1

𝑚𝑥 −

𝑏

𝑚; slope:

1

𝑚; y-

intercept: −𝑏

𝑚

15. ℓ = (𝑣

1.34)

2; about 31.3 ft

16. 8𝑥1

3

17. 3

49𝑥4

18. 12√𝑥5

19. √𝑥76

20.


Page 9 of 10

6.5

1. ; D: 𝑥 ≥ 0; R: 𝑦 ≤ 0

2. ; D: 𝑥 ≥ 0; R: 𝑦 ≤ 0

3. ; D: 𝑥 ≥ 0; R: 𝑦 ≥ 0 4. D

5. ; D: ℝ; R: ℝ

6. ; D: ℝ; R: ℝ

7. ; D: ℝ; R: ℝ

8. ; D: 𝑥 ≥ −1; R: 𝑦 ≥ 8

9. ; D: ℝ; R: ℝ

10. ; D: ℝ; R: ℝ

11. ; D: ℝ; R: ℝ 12. The domain is limited because the square root of a negative number is not a real number. The range is limited because the square root of a number is nonnegative.

13. 𝑦 = 3√𝑥 + 23

14. 𝑣 = 331.5√1 +𝐶

273.15;

D: 𝐶 ≥ −273.15; R: 𝑣 ≥ 0

15. 𝑓−1(𝑥) =5−𝑥

2

16. 𝑓−1(𝑥) = −√2𝑥

3

4

17. 3

2𝑥−7

18. 7𝑥2√𝑥 19. 2, 3 20. 𝑥 > 6

6.6 1. extraneous 2. 1 3. 29 4. 343 5. 8 6. -37 7. ±32 8. 108 9. Both sides must be raised to the power; x = 18 10. 3, 8

11. 1

2

12. 3

13. 1

4

14. (4, 25) 15. about 391 min

16. 17.

18. Show work of finding 𝑓(𝑔(𝑥)) and

𝑔(𝑓(𝑥))

19. 20𝑥7

6 20. -5


Page 10 of 10

6.Review 1. 3.50 2. 0.98 ft 3. 𝑞3

4. 𝑥4

3

5. 5√33

6. 2𝑥𝑦2 √2𝑥35

7. −𝑥2 + 𝑥 + 2 8. 𝑥3 + 2𝑥2 9. 𝑥2 + 4𝑥 + 4

10. 𝑦 =√𝑥3

4

11. 𝑦 = − √𝑥 + 210

12. 𝑦 = √𝑥

2

4

13. D: x ≥ 0; R: y ≥ 0

14. D: All real; R: All real

15. D: All real; R: All real

16.D: x ≥ 2; R: y ≥ -3

17. 98

18. 31

3

19. 5

20. −1+√37

2 (

−1−√37

2 is extraneous)

21. No Solution (-4 is extraneous)

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Algebra 2 6 Rational Exponents and Radical Functions ...rwright/algebra2/homework/Chapter 06 R… · Algebra 2 6 Rational Exponents and Radical Functions Practice Problems Page 4