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