Download 12-5A Perfect Cubes and Cube Roots

12-5A Perfect Cubes and Cube Roots
Name
Objective
To find perfect cubes and evaluate cube roots of small perfect cubes
Problem 1: A gift box is shaped like a cube. Its edges are 3 in. long.
What is the volume of the box?
! To find the volume of the box, use the formula for the volume of prisms.
A cube is a prism with all edges the same length.
V ! Bh
Formula for the volume of a prism
2
V!e "e
Let e be the length of an edge.
3
V!e
Formula for the volume of a cube
Copyright © by William H. Sadlier, Inc. Permission to duplicate classroom quantities granted to users of Foundations of Algebra.
Substitute 3 for e in the formula.
33
V!
V!3"3"3
V ! 27
Key Concept
Perfect Cubes
• To cube a number, use it
as a factor three times.
a3 ! a " a " a
So, the volume of the gift box is 27 in.3
• A perfect cube is the
Since 27 = 33, 27 is a perfect cube.
Problem 2: A recycling bin is shaped like a cube. The volume of the bin
is 64 ft3. What is the length of each edge?
! To find each edge length, use what you know about the volume of a
cube. You cube the edge length to find the volume of a cube, so to find
the length of an edge, you can find the cube root of the volume.
Make a table or use prime factorization to find a cube root.
Method 1 Make a table.
product of three equal
factors.
Key Concept
Cube Roots
A cube root is one of three
equal factors of a number.
Cube Root
Number Cubed
Perfect Cube
1
1•1•1
1
• Look for 64 in the Perfect Cube column. Go to the
first column in that row to find the cube root of 64.
2
2•2•2
8
3
3•3•3
27
• 4 is the cube root of 64.
4
4•4•4
64
• List some numbers as cube roots. Cube them to
make perfect cubes.
Method 2 Use prime factorization.
• Write the prime factorization of 64.
64 ! 2 • 2 • 2 • 2 • 2 • 2
• Rewrite as the product of 3 equal factors.
64 ! (2 • 2) • (2 • 2) • (2 • 2)
• 4 is the cube root of 64 because 4 is one of three equal factors of 64.
64 ! 4 • 4 • 4
So, the length of each edge of the recycling bin is 4 ft.
Since 43 ! 64, 4 is a cube root.
Use a table and prime factorization to find the cube root of each number.
1. 216
2. 343
3. Discuss and Write Can you find the cube root of a negative perfect cube?
Give an example to justify your reasoning.
Use after SourceBook Lesson 12-5.
Chapter 12, Lesson 5A
1
12-5A Perfect Cubes and Cube Roots
Name
Practice
4. Complete the table.
Number
3
4
5
6
7
8
9
10
Number
Cubed
Find the perfect cube of the number.
5. –11
6. 12
7. –13
8. 14
Find the cube root of the number.
9. 512
10. 125
11. 2744
12. 1728
13. 1331
15. #343
16. #729
14. 2197
Solve. Be ready to explain your strategy.
17. A box in the shape of a cube has a volume of
216 in.3 The box is open at the top. What is the
area of the bottom of the box?
18. Mia has a cube-shaped box with 7-in. edges. She
has 350 cubes with 1-in. edges. How many of the
cubes will not fit in the box?
19. A shed in the shape of a cube has a volume of
512 ft3. One side has an opening that is 6 ft by
4 ft. What fraction of the area of that side of
the shed is the opening?
20. A cube-shaped cat hideaway has a volume of
8 ft3. On 3 sides are openings that are 1 ft by
1 ft. What fraction of the surface area of the
hideaway is not openings?
21. Randy said, “I am thinking of 2 positive
integers. If you add the square of one and the
cube of the other, the sum is 100.” What are
the two integers Randy is thinking of?
22. Rosa said, “The sum of the square of a number
and the cube of that number is equal to 10 times
the square of the number.” What number is Rosa
describing?
CRITICAL THINKING
23. A cube with a volume of 729 in.3 is constructed from 27 cubes, some black
and some white. A black cube is in the center of every layer and the center
of every face. What is the volume of the black cubes? Explain your strategy
for solving the problem.
2
Chapter 12, Lesson 5A
Use after SourceBook Lesson 12-5.
Copyright © by William H. Sadlier, Inc. Permission to duplicate classroom quantities granted to users of Foundations of Algebra.
Perfect
Cube