$ \sqrt{45} = \color{red}{\sqrt{9}} \sqrt{5} = \color{red}{?} $

How to Simplify Radicals

Video Tutorial with practice problems

Some Necessary Vocabulary The radicand refers to the number under the radical sign. In the radical below, the radicand is the number '5'.
Picture of radicand

Video in How To Simplify Radicals

Some Necessary Background Knowledg

I. Know your Perfect Squares!

Before you learn how to simplify radicals,you need to be familiar with what a perfect square is. Also, you should be able to create a list of the first several perfect squares. This is easy to do by just multiplying numbers by themselves as shown in the table below.

  List Perfect Squares
2*2 4
3*3 9
4*4 16
5*5 25
6*6 36
7*7 49
8*8 64
9*9 81
10*10 100
11*11 121
12*12 144
13*13 169

II. You can rewrite a radical as the product of two radical factors of its radicand !

That's a very fancy way of saying that you can rewrite radicals as shown in the table below

Original Radical Radical rewritten as product of factors
radical 8

How to Simplify Radicals Steps

Let's look at radical 8 to help us understand the steps involving in simplifying radicals.

Step 1

Find the largest perfect square that is a factor of the radicand

4 is the largest perfect square that is a factor of 8

Step 2

Rewrite the radical as a product of the square root of 4 (found in last step) and its matching factor(2)

Step 3

Simplify

Simplify the radicals below

Problem 1

Simplify radical 72

Follow the steps for simplifying radicals.

Step 1

Find the largest perfect square that is a factor of the radicand (72)

36 is the largest perfect square that is a factor of 72

Step 2

Rewrite the radical as a product of the square root of 36 (found in last step) and its matching factor (2)

Step 3

Simplify

Problem 2

Simplify radical 72

Follow the steps for simplifying radicals.

Step 1

Find the largest perfect square that is a factor of the radicand (50)

25 is the largest perfect square that is a factor of 50

Step 2

Rewrite the radical as a product of the square root of 25 (found in last step) and its matching factor (2)

Step 3

Simplify

Problem 3

Simplify radical 72

You know the deal. Just follow the steps

Step 1

Find the largest perfect square that is a factor of the radicand (75)

25 is the largest perfect square that is a factor of 75

Step 2

Rewrite the radical as a product of the square root of 25 (found in last step) and its matching factor (3)

Step 3

Simplify

Problem 4

Simplify radical 72

Follow the steps for simplifying radicals

Step 1

Find the largest perfect square that is a factor of the radicand (32)

16 is the largest perfect square that is a factor of 32

Step 2

Rewrite the radical as a product of the square root of 16 (found in last step) and its matching factor (2)

Step 3

Simplify

Problem 5

Simplify radical 72

Hopefully, by know you know how to simplify radicals

Step 1

Find the largest perfect square that is a factor of the radicand (200)

100 is the largest perfect square that is a factor of 200

Step 2

Rewrite the radical as a product of the square root of 100 (found in last step) and its matching factor (2)

Step 3

Simplify

Problem 6

Simplify radical 72

Remember just follow the steps for how to simplify radicals

Step 1

Find the largest perfect square that is a factor of the radicand (108)

36 is the largest perfect square that is a factor of 108

Step 2

Rewrite the radical as a product of the square root of 108 (found in last step) and its matching factor (3)

Step 3

Simplify

$$ \color{red}{\sqrt{36}} \sqrt{3} \\ \color{red}{ 6} \sqrt{3} $$
Problem 7

Simplify radical 72

Ok, this question is a trick one to see if you really understand step 1 of how to simplify radicals

radical 26 cannot be simplified because this radicand (26) does not have any perfect square factors. Therefore, you cannot simplify it.

How to Simplify Radicals with Coefficients

Let's look at radical 8 to help us understand the steps involving in simplifying radicals that have coefficients. All that you have to do is simplify the radical like normal and, at the end, multiply the coefficient by any numbers that 'got out' of the square root.

Step 1

Find the largest perfect square that is a factor of the radicand (just like before)

4 is the largest perfect square that is a factor of 8

Step 2

Rewrite the radical as a product of the square root of 4 (found in last step) and its matching factor(2)

Step 3
Simplify
Step 4

Multiply original coefficient (3) by the 'number that got out of the square root ' (2)

Practice Simplifying Radicals with Coefficients

Problem 8

Simplify radical 72

Follow the steps for simplifying radicals with coefficients

Step 1

Find the largest perfect square that is a factor of the radicand (just like before)

4 is the largest perfect square that is a factor of 20

Step 2

Rewrite the radical as a product of the square root of 4 (found in last step) and its matching factor(5)

Step 3

Simplify

Step 4

Multiply original coefficient (6) by the 'number that got out of the square root ' (2)

Problem 9

Simplify radical 72

Follow the steps for simplifying radicals with coefficients

Step 1

Find the largest perfect square that is a factor of the radicand (just like before)

16 is the largest perfect square that is a factor of 80

Step 2

Rewrite the radical as a product of the square root of 16 (found in last step) and its matching factor(5)

Step 3

Simplify

Step 4

Multiply original coefficient (2) by the 'number that got out of the square root ' (2)

Problem 10

Simplify radical 72

Follow the steps for simplifying radicals with coefficients

Step 1

Find the largest perfect square that is a factor of the radicand (just like before)

25 is the largest perfect square that is a factor of 125

Step 2

Rewrite the radical as a product of the square root of 25 (found in last step) and its matching factor(5)

Step 3

Simplify

Step 4

Multiply original coefficient (4) by the 'number that got out of the square root ' (5)