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Perfect Squares

Radicals (square roots)

√4 = 2

√9 = 3

√16 = 4

√25 =5

√36 =6

√49 = 7

√64 =8

√81 =9

√100 =10

Your radical is in the simplest form when the radicand cannot be divided evenly by a perfect square.

Radicals ( or roots ) are the opposite of exponents. For instance, 3 squared equals 9, but if you take the square root of nine it is 3.

For example

4^2=16 and √16 =4

2^3=8 and ∛8 = 2

3^3= 27 and ∛27=3

The check mark √ is called the radical symbol and gives you three pieces of important information, the degree, the radical symbol, and the radicand.

The Degree is the number of times the number ( called the radicand) has been multiplied by itself.

A degree of 3 is a cube root, a 4 is the fourth root. Most times for a square root a degree will be missing, and it is assumed that it is a square root.

Method 1. Find the largest perfect square that will divide into the radical

Simplify √32

Step 1 Find the largest perfect square that will divide evenly into 32

16 * 2 =32

Step 2 Write these numbers under the radical √(16*2)

Step 3 Give each number it’s own radical sign √16*√2

Step 4 “Take out” or reduce the perfect square 4√2 ( perfect square of 16=4)

Simplify √75

25*3 =75 Step 1. Find largest perfect square

√(25*3) Step 2. Write the numbers under the radical.

√25*√3 Step 3. Give the numbers a radical sign.

5√3 Step. 4. Take out any perfect squares (square of 25=5)

Method 2 Simplify a radical " Jailbreak Method"

I have taught this method to my students for many years it it helps many remember the steps of simplifying

Simplify √75

Step 1 Create a factor tree

Step 2 When the radical is a square root any like pair of numbers escape from under the radical. In this example the pair of 5’s escape and the 3 remains under the radical.

√(5 5 3) the 5’s jailbreak and escape in a pair and the three remains under the radical

5√3

Simplify √96

Step 1. Draw a factor tree

Step 2 Pairs of like numbers escape In this example you have 2 pairs of 2

Step 3 Multiply the pairs together 2*2 = 4√(2 3)

Step 4 Multiply the numbers remaining under the radical 2*3 = 6

In order to simplify a square root you take out anything that is a perfect square.

Being familiar with the following list of perfect squares will help when simplifying radicals.

Radicand The value that you are taking the root of.

Radical symbol. The symbol that looks like a check mark. √ The bar tells you how much of the value to use. You can think of the bar as parenthesis. For example,

The bar is over 30 -3 so subtract 30 -3 =27,then take the cube root.

The bar is only over the 27 so take the cube root of 27 which equals 3 then subtract 2

Perfect Squares

4 = 2x2

9= 3x3

16= 4x4

25= 5 x5

36= 6x6

49= 7x7

64= 8x8

81= 9x9

100=10 x 10

Memorizing perfect cubes will also help when simplifying radicals

8 = 2 x 2 x 2 125 = 5 x 5 x 5 512 8 x 8 x 8

27= 3 x 3 x3 216 = 6 x 6 x 6 729 = 9 x 9 x 9

64 = 4 x 4 x 4 343 = 7 x 7 x 7

The Product Rule of Radicals states that the product of two