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
Algebra 2 6 Rational Exponents and Radical Functions Practice Problems
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.
Algebra 2 6 Rational Exponents and Radical Functions Practice Problems
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)
Algebra 2 6 Rational Exponents and Radical Functions Practice Problems
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
Algebra 2 6 Rational Exponents and Radical Functions Practice Problems
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.
Algebra 2 6 Rational Exponents and Radical Functions Practice Problems
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
Algebra 2 6 Rational Exponents and Radical Functions Practice Problems
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
Algebra 2 6 Rational Exponents and Radical Functions Practice Problems
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.
Algebra 2 6 Rational Exponents and Radical Functions Practice Problems
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
Algebra 2 6 Rational Exponents and Radical Functions Practice Problems
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)