Download 1.5 Cubes and Cube Roots

1.5
Cubes and Cube Roots
Lesson Objectives
Vocabulary
• Find a cube of a number.
cube (of a number)
cube root
• Find a cube root of a perfect cube.
perfect cube
Le
• Evaluate numerical expressions involving whole number
exponents.
arn Find a cube of a whole number.
a) A cube has edges 2 centimeters long. Find its volume.
2 cm
2 cm
2 cm
Volume of cube5 2 3 2 3 2
5 8 cm3
2 3 2 3 2 is called the cube of 2.
23 is read as “2 cubed”.
You can write 2 3 2 3 2 as 23.
So, 23 5 8.
The number 3 in 23 is the exponent. The number 2 is the base.
The cube of a whole number is called a perfect cube.
Since 8 5 2 3 2 3 2, 8 is a perfect cube.
b)
Find the cube of 7.
73 5 7 3 7 3 7
5 343
I can relate this to finding the
volume of a cube with edges
of length 7 units.
The cube of 7 is 343.
7 units
7 units
7 units
Lesson 1.5 Cubes and Cube Roots
33
Guided Practice
Find the cube of each number.
Le
1 5
2 6
3 9
arn Find a cube root of a perfect cube.
a)
A cube has a volume of 27 cubic meters. Find the length of each edge of
the cube.
?
Volume 5 27 m3
You know that
Volume of cube 5 edge 3 edge 3 edge.
To find the length of the edge of the cube, you need to find a number whose
cube is 27.
You know that
3 3 3 3 3 5 27.
So, the length of each edge of the cube is 3 meters.
3 is called the cube root of 27. This can be written as
3
27 5 3.
Finding the cube root
of a number is the
inverse of finding the
cube of a number.
34
Chapter 1 Positive Numbers and the Number Line
b)
Find the cube root of 64.
By prime factorization,
64
I can relate this to finding the length
2
of an edge of a cube, when I know it
32
·
has a volume of 64 units3.
2
2
·
2
·
16
·
2
·
2
·
?
8
Volume 5 64 units3
2
·
2
·
2
·
2
·
2
·
2
·
2
·
2
·
4
2 ·
2
64
5 2 · 2 · 2 · 2 · 2 · 2
5 (2 · 2) · (2 · 2) · (2 · 2)
5 (2 · 2)3
5 43
So,
3
Write the prime factorization.
Apply the Commutative Property of Multiplication.
Rewrite using an exponent.
64 5 4.
Guided Practice
Find the cube root of each number.
Le
4 216
5 343
6 1,000
arn Evaluate numerical expressions that contain exponents.
In order to evaluate expressions with exponents, you need to follow the order of
operations.
Order of Operations
STEP 1
Evaluate exponents.
STEP 2
Evaluate inside parentheses.
STEP 3
Multiply and divide from left to right.
STEP 4
Add and subtract from left to right.
Continue on next page
Lesson 1.5 Cubes and Cube Roots
35
a)
Find the value of 32 1 42.
32 5 3 3 3
5 9
42 5 4 3 4
5 16
Caution
32 1 42 does not have the same value
as (3 1 4)2.
So, 32 1 42 5 9 1 16
5 25
b)
Find the value of 72 3 22 1 33.
72 3 22 1 33 5 49 3 4 1 27
5196 1 27
5 223
c)
Evaluate terms with exponents first.
Then add.
Evaluate terms with exponents first.
Then multiply.
Finally, add.
Find the value of 103 2 42 3 52.
103 2 42 3 52 5 1,000 2 16 3 25
5 1,000 2 400
5 600
Evaluate terms with exponents first.
Then multiply.
Finally, subtract.
Guided Practice
Complete.
7 Find the values of 52 1 53 and 5 · 5 · 5 · 5 · 5.
525
?
3
? 5 · 5 · 5 · 5 · 5 5
3
?
5 ?
535
?
3
?
5 ?
So, 52 1 535
5
?
1
?
? Find the value of each of the following.
8 63 1 42
9 73 2 43
10 32 3 53 1 92
11 83 4 42 2 52
12 72 1 63 4 23
13 93 2 42 3 33
36
Chapter 1 Positive Numbers and the Number Line
? Practice 1.5
Find the cube of each number.
1 8
2 3
3 10
5 512
6 729
Find the cube root of each number.
4 125
Solve.
7 List the perfect cubes that are between 100 and 600.
8 Find the value of each expression. Then describe any
patterns you see.
a) 22 2 12
b) 32 2 22
c) 42 2 32
d) 52 2 42
9 Find two consecutive numbers whose squares differ by 17.
Find the value of each of the following.
10 83 1 52
11 103 2 62
12 33 3 92
13 73 4 32
14 72 1 83 2 42
15 93 2 52 1 62
16 83 3 53 4 52
17 103 4 82 3 42
18 73 2 102 4 22
19 33 1 43 3 62
Find the value of each of the following.
20 173
21 163
22 183
3
23 1, 728 3
24 8, 000 3
25 3, 375
Solve.
26 Given that 113 5 1,331, find the cube of 110.
27 Given that 143 5 2,744, find the cube root of 2,744,000.
Lesson 1.5 Cubes and Cube Roots
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