3.2 Square Roots and Cube Roots

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FCP 10
3.2 Square Roots and Cube Roots
Study the table below of perfect squares and perfect cubes. What patterns do you
observe?
Perfect Prime
Squares Factorization
1
Square
Root
1 1
Perfect
Cubes
Prime Factorization
1
Cube Root
3
1
1
2
4
2x2
4
2
8
2x2x2
3
8
9
3x3
9
3
27
3x3x3
3
27
3
16
2x2x2x2
16
4
64
2x2x2x2x2x2
3
64
4
25
5x5
25
5
125
5x5x5
3
125
5
36
2x2x3x3
36
6
216
2x2x2x3x3x3
3
216
6
100
2x2x5x5
100
10
1000
2x2x2x5x5x5
3
1000
10
144
2x2x2x2x3x3
144
12
1728
2x2x2x2x2x2x3x3x3
3
1728
12
49
Thus
And
3
729
______ because ____________
________ because ____________
A perfect square is the product of _________ equal _______________
A perfect cube is the product of __________ equal ________________
How can prime factorization be used to determine if a number is a perfect square?
How can prime factorization be used to determine if a number is a perfect cube?
Notice that some numbers are both perfect squares and perfect cubes (1 and 8)
1. Which of the following numbers is a perfect square? A perfect cube? Neither?
Justify your answer using i) prime factorization ii) your calculator
a) 512
b) 300
c) 729
Prime Factorization
a) 512
b) 300
c) 729
300
729
Calculator
512
3
512
3
300
3
729
2. Determine the answers to the following: (Use a calculator only when appropriate)
a)
4
2
b)
d)
5
e)
g)
25x 2
h)
3
4
3
5
3
3
16 25
c)
32
f)
27
81
i)
36
25
j)
m)
3
27
8
k)
3
6
8
n)
3
3
27 125
27d 3
l)
o)
3
125x 3
3
27 000
3. The volume of a cube is 512 in3. What is the surface area of the cube?
4. The surface area of a sugar cube is 13.5 cm2. What is the volume of the cube?
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