How to Add Square Roots

Examples, formula and 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 Add Square Roots

How to Add Simplified Square Roots?

Let's look at the following exampleradical 8

You can only add square roots (or radicals) that have the same radicand. So in the example above you can add the first and the last terms:

The same rule goes for subtracting. Consider the following example:

You can subtract square roots with the same radicand--which is the first and last terms.

Practice Problems

Directions:Add the square roots below

Problem 1

Only the first and last square root have the same radicand, so you can add these two terms.


Problem 2

Remember--the same rule applies to subtracting square roots--the radicands must be the same.

How to Add Square Roots- that are not simplified

Let's look at the following exampleradical 8

You may immediately see the problem here: The radicands are not the same. Therefore, we can not add them at the moment. However, if we simplify the square roots first, we will be able to add them. Let's use this example problem to illustrate the general steps for adding square roots.

Step 1 Step 1
Add square roots with the same radicands.
Problem 3
Step 1
Simplify each square root
Step 2

Add square roots with the same radicands

Problem 4
Step 1
Step 2

Add square roots with the same radicands

Problem 5
Step 1
Step 2

Add square roots with the same radicands

Problem 6
Step 1

Simplify each square root

Step 2

Remember--the same rule applies to subtracting square roots with the same radicands

How to Add Square Roots with coefficients?

Let's look at the following exampleradical 8

Do you see what distinguishes this expression from the last several problems?

Each square root has a coefficent. The rules for adding square roots with coefficients are very similar to what we just practiced in the last several problems--with 1 additional step --which is to multiply the coefficeints with the simplified square root.

Step 1
Ignore the coefficients ( 2 and 5) and simplify each square root.
Step 1
Multiply the coefficients (2 and 5) by any numbers that 'got out' of the square root (3 and 2, respectively).
Step 1
Add any radicals with the same radicand
Problem 7
Step 1

Ignore the coefficients ( 4 and 5) and simplify each square root.

Step 2

Multiply the coefficients (4 and 5) by any numbers that 'got out' of the square root (3 and 2, respectively).

Step 3

Add any radicals with the same radicand