1. Square Roots and Cubes Roots of Whole Numbers
Tammy Wallace

2. Perfect Squares
A number is a perfect square if it is the product of a number and itself. The first 12 perfect squares:
NUMBER MULTIPLIED
BY ITSELF
PERFECT
SQUARES
1 x 1 =
2 x 2 =
3 x 3 =
4 x 4 =
5 x 5 =
6 x 6 =
NUMBER MULTIPLIED
BY ITSELF
PERFECT
SQUARES
7 x 7 =
8 X 8 =
9 X 9 =
10 x 10 =
11 x 11 =
12 x 12 = 1 49 4 64 9 81 16
100 25
121 36
144

3. Square Roots
Taking the square root of a number is the inverse operation to squaring a number (or multiplying a number by itself). For example if 3 2 = _________, then 9 =_________. The symbol tells you to 9 3
find the number that multiplies by itself to give you the inside number.

4. Parts of a Radical
An expression that contains a square root is called a radical. It can have three parts.
Radicand: the value you are taking the ‘root’ of
Index: tells you which root you are looking for; if there is not index, it is assumed to be 2.
Coefficient: it is being multiplied by the radical.

5. Simplify
100 = 25 = 144 = 𝟏𝟎 𝟓 𝟏𝟐

6. Non-Perfect Radicals
Simplify: 24 Since 24 is not a perfect square, its square root is not a whole number. To simplify this radical, 24 needs to be rewritten as the product of two factors. However, one of the factors must be a perfect square.

7. Simplify 24
What is the highest factor of 24 that is also is a perfect square? 4
Therefore, 24 = ____ X _____ 24
= 4∙6
= 4 ∙ 6
=𝟐 𝟔 4 6

8. Simplify 32
What is the highest factor of 32 that is also is a perfect square? 16
Therefore, 32 = ____ X _____ 32
= 16∙2
= 16 ∙ 2
=𝟒 𝟐 16 2

9. Simplify 54
What is the highest factor of 54 that is also is a perfect square? 9
Therefore, 54 = ____ X _____ 54
= 9∙6
= 9 ∙ 6
=𝟑 𝟔 9 6

10. Perfect Cubes
A number is a perfect cube if it is the product of a number and itself three times. The first 5 perfect cubes:
NUMBER MULTIPLIED
BY ITSELF 3 TIMES
PERFECT
CUBES
1 x 1 x 1 =
2 x 2 x 2 =
3 x 3 x 3 =
4 x 4 x 4 =
5 x 5 x 5 = 1 8 27 64
125

11. Cube Roots
Taking the cube root of a number is the inverse operation of cubing a number. For example if 2 3 = _________, then 3 8 =_________. So, the symbol 3 tells you to find the number that multiplies by itself three times to give you the inside number – the radicand. 8 2

12. Simplifying the following Radical Expressions
3 125 = 3 27 = 3 64= 5 3 𝟒

13. Non-Perfect Cubes =3 3 2 Simplify 3 54 2 27
What is the highest factor of that is also is a perfect cube? 𝟐𝟕
Therefore, 54 = ____ X _____
3 54
= 3 27∙2
= 3 27 ∙ 3 2
=3 3 2 27 2

14. Simplify 3 24
What is the highest factor of 24 that is also is a perfect cube? 𝟖
Therefore, 54 = ____ X _____
3 24
= 3 8∙3
= 3 8 ∙ 3 3
=2 3 3 𝟖 𝟑

15. Simplify
What is the highest factor of 128 that is also is a perfect cube? 64
Therefore, 54 = ____ X _____
3 128
= 3 64∙2
= 3 64 ∙ 3 2
=4 3 2 𝟔𝟒 𝟐