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

**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 |
---|---|

### How to Simplify Radicals Steps

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

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

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

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

Simplify

### Simplify the radicals below

Follow the steps for simplifying radicals.

Step 1Find the largest perfect square that is a factor of the radicand (72)

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

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

Simplify

Follow the steps for simplifying radicals.

Step 1Find the largest perfect square that is a factor of the radicand (50)

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

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

Simplify

You know the deal. Just follow the steps

Step 1Find the largest perfect square that is a factor of the radicand (75)

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

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

Simplify

Follow the steps for simplifying radicals

Step 1Find the largest perfect square that is a factor of the radicand (32)

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

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

Simplify

Hopefully, by know you know how to simplify radicals

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

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

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

Simplify

Remember just follow the steps for how to simplify radicals

Step 1Find the largest perfect square that is a factor of the radicand (108)

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

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

Simplify

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 1Find 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

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

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

**Practice** Simplifying Radicals with Coefficients

Follow the steps for simplifying radicals with coefficients

Step 1Find 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

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

Simplify

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

Follow the steps for simplifying radicals with coefficients

Step 1Find 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

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

Simplify

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

Follow the steps for simplifying radicals with coefficients

Step 1Find 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

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

Simplify

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