HW #46
Algebra 2
Name: _____________________________
Date: ___________________ Period: ____
nth roots and rational exponents
1.
Rewrite the expression using rational exponent notation.
3
2
7
b. √11
a. ( √5)
4
d. √14
2.
9
5
Rewrite the expression using radical notation.
1
1
b. 74
2
3
c. 107
4.
11
f. √8
e. ( √16)
a. 63
3.
8
c. ( √2)
d. 55
Find the indicated real nth root(s) of a.
a. 𝑛 = 2, 𝑎 = 100
b. 𝑛 = 4, 𝑎 = 0
c. 𝑛 = 3, 𝑎 = −8
d. 𝑛 = 7, 𝑎 = 128
e. 𝑛 = 6, 𝑎 = −1
f. 𝑛 = 5, 𝑎 = 0
Evaluate the expression without using a calculator.
3
3
a. √64
b. √−1000
6
c. − √64
4. Evaluate the expression without using a calculator.
−1
1
d. 4 2
4
e.
2
3
i. ( √0)
4
−3
−(256)4
6
h. ( √−27)
−2
k. 325
j. − (25 2 )
5.
f.
−4
3
g. ( √16)
1
13
l. (−125) 3
Solve the equation. Round your answer to two decimal places when appropriate.
b. 6𝑥 3 = −1296
a. 𝑥 5 = 243
c. 𝑥 6 + 10 = 10
d. (𝑥 − 4)4 = 81
f. (𝑥 + 12)3 = 21
e. −12𝑥 4 = −48
EXAM REVIEW
1. Write the radical expression
exponential notation.
 7
3
4
in
2. Simplify the radical expression
A. 6
A. 71
B. 2 3 9
B. 712
C. 6 3 6
C. 7
3
4
4
D. 7 3
D. 72
3
216 .
HW #47
Algebra 2
Name: _____________________________
Date: ___________________ Period: ____
Properties of Rational Exponents
1.
Simplify the expression.
1 6
1
a. (24 ∙ 23 )
1
d.
52
703
e. ( 2)
1
14 3
3
g. √8 ∙ √2
2.
c.
1
−1
36 2
−1
2
3
f. √64 ∙ √64
4
12
5
i √1215
h. ( √6 ∙ √6)
Perform the indicated operation.
5
5
a. √6 + 5 √6
8
1
8
4
4
1
b. 44 ∙ 644
8
c. − √4 + 5 √4
1
1
b. 5(5)7 − 7(5)7
1
1
d. 1602 − 102
3.
Simplify the expression. Assume all variables are positive.
3
5
a. √32𝑥 5
b.
𝑥7
1
𝑥3
4
d. √10𝑥 5 𝑦 8 𝑧10
4 𝑥 12
𝑦4
c. √
5
5
e. √8𝑥𝑦 7 ∙ √6𝑥 6
9𝑥 2 𝑦
f. √
32𝑧 3
3
5
√𝑥 3
g. 7
h.
√𝑥 4
−1
𝑥 4 𝑦𝑧 3
1 2
𝑥3𝑧 3
EXAM REVIEW
1
4
1
12
1. What is the value of the expression 64  64 ?
A. 2
B. 4
C. 8
D. 16
2. Simplify the expression:
 32 3 
 3x y 


A. 3xy 6
B. 6x 3 y 9
C. 9x 3 y 6
D. 9xy 9
2
HW #48
Algebra 2
Name: _____________________________
Date: ___________________ Period: ____
Power Functions and Function Operations
2
1.
c.
2.
1
Let 𝑓(𝑥) = 2𝑥 3 and 𝑔(𝑥) = 3𝑥 2 . Perform the indicated operation and state the domain.
a. 𝑓(𝑥) ∙ 𝑔(𝑥)
b. 𝑓(𝑥) ∙ 𝑓(𝑥)
𝑓(𝑥)
𝑔(𝑥)
d.
𝑔(𝑥)
𝑔(𝑥)
Let 𝑓(𝑥) = 10𝑥 and 𝑔(𝑥) = 𝑥 + 4. Perform the indicated operation and state the domain.
a. 𝑓(𝑥) + 𝑔(𝑥)
b. 𝑓(𝑥) − 𝑔(𝑥)
c. 𝑓(𝑥) ∙ 𝑔(𝑥)
d.
𝑓(𝑥)
𝑔(𝑥)
e. 𝑓(𝑔(𝑥))
f. 𝑔(𝑓(𝑥))
g. 𝑓(𝑓(𝑥))
h. 𝑔(𝑔(𝑥))
3.
Perform the indicated operation and state the domain.
1
𝑓
a. ; 𝑓(𝑥) = 9𝑥 −1 , 𝑔(𝑥) = 𝑥 4
𝑔
1
c. 𝑓(𝑓(𝑥)); 𝑓(𝑥) = 2𝑥 5
𝑓
b. ; 𝑓(𝑥) = 𝑥 2 − 5𝑥, 𝑔(𝑥) = 𝑥
𝑔
d. 𝑓(𝑔(𝑥)); 𝑓(𝑥) = 6𝑥 −1 , 𝑔(𝑥) = 5𝑥 − 2
e. 𝑔(𝑓(𝑥)); 𝑓(𝑥) = 𝑥 2 − 3, 𝑔(𝑥) = 𝑥 2 + 1
EXAM REVIEW
1. Let f  x   x  2 and g  x   x 2  3 x . What
expression is equal to f  g  x   ?
A. x 2  3 x  2
B. x 2  x  2
2. If f  x   x 2  x  1 and g  x   3 x  6 , what
is the
product of f  1 and g  3  ?
A. −9
B. −3
C. x  x  6 x
C. 9
D. x  4 x  2
D. 0
3
2
2
HW #49
Algebra 2
Graphing Square Root and Cube Root Functions
1.
Name: _____________________________
Date: ___________________ Period: ____
Graph the function. State the domain and range.
1
b. 𝑓(𝑥) = √𝑥 + 6
a. 𝑦 = 𝑥 2 − 2
1
c. 𝑔(𝑥) = −(𝑥 − 7)2
3
e. ℎ(𝑥) = √𝑥 − 7
1
d. 𝑦 = (𝑥 − 1)2 + 7
3
f. 𝑦 = √𝑥 − 5
1
1
g. 𝑦 = −(𝑥 − 2)3 + 3
h. 𝑦 = (𝑥 + 1)3 − 2
i. 𝑔(𝑥) = 4√𝑥 + 8
j. 𝑓(𝑥) = 0.5√𝑥 + 2
EXAM REVIEW
1. What is the graph of y 
A
B
x 3?
C
D
HW #50
Algebra 2
Name: _____________________________
Date: ___________________ Period: ____
Piecewise Functions
1.
Graph each function.
 x  1 x  0
2 x  1 x  0
a. f ( x )  
 x  4

c. f ( x)   2

 x2
x  1
1  x  1
x 1
 x  3 x  1
2
 2 x  2 x  1
b. f ( x )  
 3x  5
x0

x
0 x3
d. f ( x)  

x3
 x3 2
2.
Write equations for the piecewise functions whose graphs are shown below. Assume that the units are one for every tick
mark.
a.
b.
EXAM REVIEW
1. What is the graph of y  x  1  5 ?
A
B
C
D
HW #51
Algebra 2
Name: _____________________________
Date: ___________________ Period: ____
Inverses
1.
Verify that f and g are inverse functions.
a. 𝑓(𝑥) = 𝑥 + 7, 𝑔(𝑥) = 𝑥 − 7
c. 𝑓(𝑥) =
e. 𝑓(𝑥) =
1
2
1
3
𝑥 + 1, 𝑔(𝑥) = 2𝑥 − 2
𝑥 2 , 𝑥 ≥ 0, 𝑔(𝑥) = (3𝑥)
1
1
3
3
b. 𝑓(𝑥) = 3𝑥 − 1, 𝑔(𝑥) = 𝑥 +
d. 𝑓(𝑥) = 3𝑥 3 + 1, 𝑔(𝑥) = (
1⁄
2
f. 𝑓(𝑥) =
𝑥 5 +2
7
5
𝑥−1
3
)
, 𝑔(𝑥) = √7𝑥 − 2
1⁄
3
2.
Find the inverse function. Be sure to indicate the domain restriction if it is needed.
b. 𝑓(𝑛) = −(𝑛 − 2)3
a. 𝑔 (𝑛) = √2𝑥 − 4 + 5
3.
f. ℎ(𝑛) = 2𝑛4 − 32 where x ≤ 0
e. 𝑓(𝑥) = 2𝑥 5
3 𝑥−1
d. 𝑔(𝑥) = √
c. ℎ(𝑥) = (𝑥 + 2)2 − 3 where x > 0
2
Find the inverse of each function. Then graph the function and its inverse.
3
1
b. 𝑓(𝑥) = √𝑥 + 2 − 2
a. 𝑔(𝑥) = 𝑥 + 4
c. 𝑔(𝑥) = √𝑥 − 3 + 2
2
EXAM REVIEW
1. Which is the inverse of the function
x 3
, where x ≥ 0
2
3x 2  3
B. y 
, where x ≥ 0
2
A. y 
y
2x  3
?
3
2. What is the inverse of the function y 
A. y  x  1
2
C. y 
9x2  3
, where x ≥ 0
2
D. y 
x2
, where x ≥ 0
2
B. y 
3
1
x3
C. y   x  3
D. y  x 3  3
3
3
x3 ?
HW #52
Algebra 2
Name: _____________________________
Date: ___________________ Period: ____
Solving Radical Equations
1. Solve the equation. Check for extraneous solutions.
1
3
2
a. 𝑥3 − = 0
5
4
c. 3(𝑥 + 1)3 = 48
3
b. 4𝑥4 = 108
3
d. √𝑥 + 10 = 16
e. √𝑥 + 40 = −5
f. 2√7𝑥 + 4 − 1 = 7
g. 𝑥 − 12 = √16𝑥
h. √8𝑥 + 1 = 𝑥 + 2
i. √−3𝑥 − 5 = 𝑥 + 3
3
k. √1 − 𝑥 2 = 𝑥 + 1
j. √9𝑥 + 90 = 𝑥 + 6
3
l. √9𝑥 2 + 22𝑥 + 8 = 𝑥 + 2
EXAM REVIEW
1. What is the value of x in the equation
3
x  16  4 ?
A. x = 20
B. x = 28
C. x = 32
D. x = 80
2. What is the value of x in the equation
5 x  12  13 ?
A.
1
125
B.
1
5
C. 25
D. 125
HW #53
Algebra 2
Applying Radical Functions
1.
Name: _____________________________
Date: ___________________ Period: ____
At an amusement park a ride called the rotor is a cylindrical room that spins around. The riders stand against the circular
wall. When the rotor reaches the necessary speed, the floor drops out and the centrifugal force keeps the riders pinned to the
wall. The model that gives the speed s (in meters per second) necessary to keep a person pinned to the wall is s  4.95 r
where r is the radius (in meters) of the rotor. Estimate the radius of a rotor that spins at a speed of 8 meters per second.
2.
The speed that a tsunami (tidal wave) can travel is modeled by the equation S  356 d where S is the speed in kilometers
per hour and d is the average depth of the water in kilometers. Solve the equation for d and find the average depth of the
water for a tsunami found to be traveling at 120 kilometers per hour.
3.
The distance, d, in miles that a person can see to the horizon can be modeled by the formula d 
3h
where h is the
2
person’s height above sea level in feet. To the nearest tenth of a mile, how far to the horizon can a person see if they are 100
feet above sea level?
2
The surface area of a cube in terms of its volume is A  6V 3 . Solve the formula for V and find the volume of a cube with a
surface area of 12 square feet.
4.
EXAM REVIEW
1. An animal population can be modeled over
2
3
time by P  t   2t  10 , where t is measured
in weeks. After how many weeks will the
population be 18 animals?
A. 8
2. A company that produces DVDs uses the
formula
1
C  90n 3  350
to calculate the cost C in dollars of producing
n DVDs per day. How many DVDs can be
produced for a cost of $800?
B. 12
A. 15
C. 23
B. 45
D. 27
C. 75
D. 125
3. Simplify the expression  2  i  3  2i  where
i  1 .
4. If f  x   3 x  1 and g  x   2 x  3 , what is
the product of f  4  and g  1 ?
A. 8 + 5i
A. –55
B. 8 – i
B. –13
C. 4 + 5i
C. –10
D. 4 – i
D. 8