10-4
Cube Root Equations
TEKS FOCUS
VOCABULARY
TEKS (6)(B) Solve cube root equations that
have real roots.
TEKS (1)(B) Use a problem-solving
model that incorporates analyzing given
information, formulating a plan or strategy,
determining a solution, justifying the
solution, and evaluating the problem-solving
process and the reasonableness of the
solution.
ĚCube root equation – A cube root equation is a radical equation in
which the radical has an index of 3. A cube root equation can also be
written using a rational exponent with a denominator of 3.
ĚFormulate – create with careful effort and purpose. You can
formulate a plan or strategy to solve a problem.
ĚReasonableness – the quality of being within the realm of common
sense or sound reasoning. The reasonableness of a solution is
whether or not the solution makes sense.
ĚStrategy – a plan or method for solving a problem
Additional TEKS (1)(A), (1)(C), (1)(D),
(7)(H)
ESSENTIAL UNDERSTANDING
Solving a cube root equation may require that you cube each side of the equation.
Problem 1
P
TEKS Process Standard (1)(B)
Solving a Cube Root Equation With Real Roots
3
What is the solution to the equation 1
5x − 1 = 4? Evaluate your
problem-solving process.
Analyze Given Information The cube root needs to be removed to solve for x.
What property allows
you to cube each side
of the equation?
If a = b, then an = b n
for any integer n.
Formulate a Plan
Solve for x algebraically.
S
11. To do so, first remove the cube root by cubing each side of the equation.
22. Solve the resulting linear equation by simplifying and combining like terms.
33. As a final step, isolate x by dividing by 5.
Determine a Solution Execute the plan to solve for x.
3
1
5x - 1 = 4
3
5x - 1)3 = (4)3
(1
5x - 1 = 64
5x = 65
x = 13
Cube each side.
Simplify.
Add 1 to each side.
Divide each side by 5.
The solution is 13.
continued on next page ▶
436
Lesson 10-4 Cube Root Equations
Problem 1
continued
Justify the Solution Check your solution in the original equation.
3
1
5x - 1 = 4
15(13) - 1 ≟ 4
Write the original equation.
3
Substitute 13 for x.
3
1
65 - 1 ≟ 4
Simplify.
164 ≟ 4
3
4=4✔
Evaluate the Problem-Solving Process The solution checks, since 4 = 4. The
problem-solving process was successful.
Problem
bl
2
Solving Equations With Rational Exponents
2
A What are the solutions of 3(x + 1)3 = 12?
An equation in which the
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reciprocal power.
2
3(x + 1)3 = 12
Why is this a cube
root equation?
You can rewrite
the equation using
radical notation as
3
2(x + 1)2 = 4.
2
(x + 1)3 = 4
2 3
3 2
Divide each side by 3.
3
2
((x + 1) ) = 4
#
2 3
2
(x + 1) 3
Raise each side to the 32 power.
3
= 42
0x + 10 = 8
Use absolute value symbols because the denominator
3
2 in the exponent indicates square root.
2
x + 1 = {8
x = 7 or x = -9
The solutions are 7 and -9.
Check
2
2
3(x + 1)3 = 12
3(x + 1)3 = 12
3(7 + 1)3 ≟ 12
3( -9 + 1)3 ≟ 12
3(23)3 ≟ 12
3(( -2)3)3 ≟ 12
3(2)2 ≟ 12
3( -2)2 ≟ 12
2
2
12 = 12 ✔
2
2
12 = 12 ✔
continued on next page ▶
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Problem 2
continued
3
Why do you isolate
the variable
expression?
If you raise each side of
3
3 2(x
+ 1)5 + 1 = 97
to the 35 power you will
end up with a more
complicated equation,
not a simpler one.
B What is the solution of 32(x + 1)5 + 1 = 97?
3
32
(x + 1)5 + 1 = 97
5
3(x + 1)3 + 1 = 97
5
3
3(x + 1) = 96
5
3
(x + 1) = 32
5 3
3 5
3
5
((x + 1) ) = 32
x+1=8
x=7
Rewrite the radical using a rational exponent.
Subtract 1 from each side.
Divide each side by 3.
Raise each side to the 35 power.
Simplify.
Subtract 1 from each side.
The solution is 7.
Problem
P
bl
3
TEKS Process Standard (1)(A)
Using a Cube Root Equation
3 V
Earth Science For Meteor Crater in Arizona, the formula d = 2 5
relates the
0.3
diameter d of the rim (in meters) to the volume V (in cubic meters). What is the
volume of Meteor Crater? (All values are approximate.)
1.2 km
What is the diameter
in meters?
1.2 km = 1.2 * 1000 m
3 V
d = 25
0.3
d
3 V
2 = 5 0.3
( d2 )3 = 0.3V
d 3
0.3 ( 2 ) = V
1200 3
0.3 ( 2 ) = V
64,800,000 = V
Solve for V. First divide each side by 2.
Cube each side.
Multiply each side by 0.3.
Substitute 1200 for d.
Simplify.
The volume of Meteor Crater is about 64,800,000 m3.
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Lesson 10-4 Cube Root Equations
Problem 4
P
TEKS Process Standard (1)(C)
Solving a Cube Root Equation by Graphing
Multiple Choice You can model the population P of Corpus Christi, Texas, between
3
the years 1970 and 2005 by the cube root function P (x) = 75,000 1x
− 1950,
where x is the year. Using this model, in what year was the population of Corpus
Christi 250,000?
1980
1983
1987
1990
3
For P = 250,000, solve the equation 250,000 = 75,000 1x - 1950.
NLINE
HO
ME
RK
O
How can you rewrite
a cube root function
using an exponent?
You can write a cube root
3
function y = 1x
as
1
y = x3.
WO
Graph
Y1 = 75000(X − 1950) ^(1/3) and Y2 = 250000.
G
Adjust the window
to find where the graphs intersect.
A
Use
U the INTERSECT feature to find the x-coordinate of the
intersection.
i
In
I the year 1987, the population of Corpus Christi was 250,000.
The correct answer is C.
PRACTICE and APPLICATION EXERCISES
Intersection
X=1987.04
Y=250000
Scan page for a Virtual Nerd™ tutorial video.
Solve each equation. Check your answer and evaluate your problem-solving process.
For additional support when
completing your homework,
go to PearsonTEXAS.com.
3
1. 1
2x + 7 = 11
3
2. 1
5 + 8x = -3
3
3. 1 = 3 + 1
4x - 8
3
4. 3 1
x + 3 = 15
3
5. 1
x+3=5
3
6. 1
2x - 1 = 3
3
7. 1
x+2-2=0
3
8. 1
2x + 3 - 7 = 0
3
9. 1
6 - 3x - 2 = 0
10. Apply Mathematics (1)(A) A diameter of a spherical water tank is 26 ft. What is the
3 6V 2
volume of the tank? 1Hint: d = 5
p
3 4V gives the diameter of a closed cylinder where V is the
11. The formula d = 5
p
volume. Boyle’s Law says that Pinitial Vinitial = Pnew Vnew . If the pressure Pinitial is
1 psi, Pnew is 8 psi and Vinitial is 2p, what is the new diameter using Boyle’s Law?
12. Use a Problem-Solving Model (1)(B) The
3 3x
function y = 5
4p relates the radius y of a
spherical gas tank (in meters) to the volume x
(in cubic meters). A company manufactures
gas tanks with a radius of 10 m. Find the
volume of the tank to the nearest cubic meter.
Use a problem-solving model by
r analyzing the given information,
r formulating a plan or strategy,
r determining a solution,
r justifying the solution, and
r evaluating the problem-solving process.
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Solve each equation.
2
13. (x + 5)3 = 4
4
15. 3(x - 2)3 = 243
1
17. 2(2x)3 + 1 = 5
4
19. 2(x - 1)3 + 4 = 36
2
14. (x + 2)3 = 9
2
16. 3 + (4 - x)3 = 12
4
18. (2x + 3)3 - 3 = 13
2
20. (x - 3)3 = x - 7
Select Tools to Solve Problems (1)(C) Solve each cube root equation by
graphing. Round the answer to the nearest hundredth, if necessary. If there
is no solution, explain why.
3
21. 1 x - 3 = 12
3
22. 1
2x - 3 = 4
3
23. 1
2x + 5 = 12 - x
3
24. 21
x = 1(x + 1)
3
25. 1(x + 3) = 41
(x) - 2
3
x - 1 = 1x - 1
26. 1
27. Suppose that a function pairs elements from set A with elements
from set B. Recall that a function is called onto if every element
in B is paired with at least one element in A.
3
a. The graph shows a transformation of y = 1
x. Write the function.
b. What are the domain and range of the function?
y
-8
-4
O
-8
c. If the domain is restricted to all real numbers greater than or equal to 10, and the
range is the set of nonnegative real numbers, is the function onto? Explain.
28. The nine-banded armadillo is a relatively recent addition to Texas. It can
jump 3–4 ft and grows to about 15–23 in. from the neck to the base of the tail. Some
armadillos roll up into a ball when frightened. The spherical shape can be used to
3
show that the function y = 1
6p 2x relates the length y of the armadillo (in inches)
to its volume x (in cubic inches). Suppose you measure the length of an armadillo
as 18 in. Write and solve a cube root equation to find the volume of the armadillo to
the nearest cubic inch.
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Lesson 10-4 Cube Root Equations
x
8
3
29. Explain how you would find the x- and y-intercepts of f (x) = 1
x + 3. Then find the
intercepts and graph the function.
30. The size of a computer case is related to the size of the motherboard, and smaller
3
cases mean that upgrading is limited. The equation s = 1
V models the length of
an edge of a computer case with volume V in cubic inches.
a. Graph the equation on your calculator.
b. Suppose you want to buy a new video card for your old case that has volume
512 in.3 . You need 0.75 inch minimum between the case and the edge of the
video card for air circulation. If video cards come in full-length 12 inches, halflength 7 inches, and low-profile 6.5 inches, which one would be the best choice?
Explain. (Video cards are installed at right angles to the sides of the case.)
TEXAS Test Practice
T
3
3
31. How is the graph of y = 1
x - 5 translated from the graph of y = 1
x?
A. shifted 5 units left
B. shifted 5 units right
C. shifted 5 units up
D. shifted 5 units down
32. Which absolute value inequality has the graph shown here?
⫺5 ⫺4 ⫺3 ⫺2 ⫺1
0
1
2
3
4
5
F. 0 x - 1 0 … 3
G. 0 x - 1 0 Ú 3
H. 0 x + 1 0 … 3
J. 0 x + 1 0 Ú 3
33. Which polynomial cannot be factored in the real number system?
A. x2 - 3x + 2
B. x2 + 4
C. 4x2 - 1
D. 2x2y - 2xy 2
34. How do the domains and ranges of f (x) = 1x - 1 and g(x) = 1x - 1 compare?
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