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Radicals
Tammy WallaceVarina High
Perfect SquaresA 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 =
14
9
16
25
36
49
6481
100
121
144
Perfect SquaresA variable is a perfect square if its exponent is EVEN.
The first 4 perfect squares variables:NUMBER
MULTIPLIED BY ITSELF
PERFECT
SQUARES𝒙𝟐
𝒙𝟒
𝒙𝟔
𝒙𝟖
Square RootsTaking the square root of a number is the inverse operation to squaring a number (or multiplying a number by itself).
For example if , then . The symbol tells you to
9 3 find the number that multiplies by itself to give you the inside number.
Parts of a RadicalAn expression that contains a square root is called a radical. It can have three parts.
ba cRadicand: the value you are taking the ‘root’ of
Coefficient: it is being multiplied by the radical.
Index: tells you which root you are looking for; if there is not index, it is assumed to be 2.
Simplify
𝟏𝟎
𝟓
𝟏𝟐
𝒙
When dealing with exponents, divide the exponent of the radicand by the index to get the exponent of the root.
Simplify𝒙𝟐
𝟐 𝒙
𝟗 𝒙𝟒𝒚
𝟔𝒂𝟑𝒃𝟐
If your radicand has more than one factor, take the root of each factor separately.
√ 4 ∙√𝑥2=¿
√81 ∙√𝑥8 ∙√ 𝑦2=¿
√36 ∙√𝑎6 ∙√𝑏4=¿
Non-Perfect RadicalsSimplify:
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.
Simplify What is the highest factor of 24 that is also is a perfect square?
4
Therefore, 24 = ____ X _____
4 6
Simplify What is the highest factor of 32 that is also is a perfect square?
16
Therefore, 32 = ____ X _____
16 2
Simplify What is the highest factor of 54 that is also is a perfect square?
9
Therefore, 54 = ____ X _____
9 6
Simplify What is the highest factor of that is also is a perfect square?
Therefore, = ____ X _____𝒙𝟒 𝒙
Simplify What is the highest factor of that is also is a perfect square?
Therefore, = ____ X _____𝟐𝟓𝒙𝟐 𝟐 𝒚
Simplify What is the highest factor of that is also is a perfect square?
Therefore, = ____ X _____𝒙𝟖 𝟒𝟐𝒙
Perfect CubesA number is a perfect cube if it is the product of a number and itself three times.
The first 5 perfect cubes: The first 3 perfect cube
variables. 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 =
VARIABLES MULTIPLIED
BY ITSELF 3 TIMES
PERFECT
CUBES
1
8
27
64
125
𝒙𝟑
𝒙𝟔
𝒙𝟗
A variable is a perfect cube if it is divisible by 3.
Cube RootsTaking the cube root of a number is the inverse operation of cubing a number.
For example if , then .
So, the symbol tells you to find the number that multiplies by itself three times to give you the inside number – the radicand.
8 2
Simplifying the following Radical
Expressions5
3
𝒙❑
𝟒 𝒙𝟒
Non-Perfect CubesSimplify
What is the highest factor of that is also is a perfect cube?
Therefore, = ____ X _____27 2
Simplify What is the highest factor of that is also is a perfect cube?
Therefore, = ____ X _____
𝟖 𝒙𝟔 𝟑 𝒙𝟐
