How to Simplify Radicals

Some Necessary Background Knowledge

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

How to Simplify Radicals Steps

Let's look at 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

Practice problem 1) Simplify

Answer

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

Answer

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

3) Simplify

Answer

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

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4) Simplify

Answer

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

5) Simplify

Answer

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

6) Simplify

Answer

Remember just follow the steps for how to simplify radicals.

Step 1) Find the largest perfect square that is a factor of the radicand (200)	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

7) Simplify

Answer

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

Problem 1) Simplify

Answer

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)

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Problem 2) Simplify

Answer

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)

3) Simplify

Answer

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)