How to Add Square Roots: Problems and Solutions

Adding Square Roots

The rules for adding together square roots are very similar to the rules for adding together variables in algebra. Just like you can't add together 'x' and 'y' unless you know what their values are, you can't add together two different square roots unless you figure out their values.

For example, you can't just add together the square root of four and the square root of nine without figuring out their values first. Since you know the square root of four is two, and the square root of nine is three, you can simplify the equation to two plus three, which equals five (2 + 3 = 5). Here are some other examples of how to add perfect squares:

• (square root of 16) + (square root of 16) = 4 + 4 = 8
• (square root of 36) + (square root of 25) = 6 + 5 = 11
• (square root of 9) + (square root of 49) = 3 + 7 = 10

Adding Imperfect Squares

How do you handle problems that ask you to add imperfect squares, like the square root of seven or the square root of ten? In many cases, it's fine to leave these square roots in your answer. For instance, (square root of two) + (square root of three) would be a perfectly acceptable answer.

Estimating Imperfect Squares

It's also possible to get approximate values for imperfect square roots using a calculator. For instance, you could find the sum of the square root of five and the square root of ten by adding their approximate values (2.2361 + 3.1622) to get 5.3983. Since including all of these decimals doesn't make your 'simplified' answer very simple, teachers will often prefer that you leave the square roots in your answer.

Adding Terms with Perfect Square Roots

If you're adding terms that have perfect squares in them, like 5(square root of 4) + 3(square root of 25), you can simplify the square roots and then multiply them by the other part of the term. In the case of 5(square root of 4), you would simplify the square root of four to two and then multiply two by five to get ten (5 x 2 = 10). For 3(square root of 25), you simplify the square root of 25 to five and then multiply it by three (5 x 3) to get 15. Last, you would add ten and 15 (10 + 15) to get the solution of 25.

Adding Terms with Imperfect Square Roots

Just as you can only add together terms with like variables, such as 3x and 4x, you can only add terms with imperfect square roots if they have the same square roots. For instance, it's fine to add together 4(square root of 2) and 5(square root of 2), since the square roots are the same. This would give you 9(square root of 2), since 4 + 5 = 9. However, you can't add together 4(square root of 2) and 4(square root of 3) because the square roots are different. Here are some more examples:

• 6(square root of 7) + 7(square root of 7) = 13(square root of 7)
• 2(square root of 6) + 9(square root of 6) = 11(square root of 6)
• 4(square root of 5) + 5(square root of 3)...cannot be added because the square roots are different.

Practice Problems

Now, you can solidify what you've learned by simplifying these expressions. Remember, if you have a perfect square, you can simplify it to an integer. Some of the expressions may already be simplified as much as they can be.

1. 2(square root of 4) + 5(square root of 16)
2. 2(square root of 11) + 8(square root of 11)
3. 10(square root of 3) + 10(square root of 7)
4. 3(square root of 6) + 9(square root of 6)
5. 2(square root of 9) + 5(square root of 9)

Solution One

For the expression 2(square root of 4) + 5(square root of 16), the correct answer is 24. The square root of four and the square root of 16 are not the same, so they cannot be added directly. However, they are both perfect squares, which means they can each be simplified to an integer and multiplied with the other part of the term they're in. Here's how it looks mathematically:

2(square root of 4) + 5(square root of 16)

= 2(2) + 5(4)

= 4 + 20

=24

Solution Two

The expression 2(square root of 11) + 8(square root of 11) can be simplified to 10(square root of 11). The square roots in both terms are the same, so you can just add two and eight to get ten (2 + 8 = 10).

Solution Three

You cannot further simplify the expression 10(square root of 3) + 10(square root of 7). The square roots are different, and they are not perfect squares that can be simplified to integers.

Solution Four

The expression 3(square root of 6) + 9(square root of 6) has two like square roots, so it can be simplified to 12(square root of 6).

Solution Five

There are two ways to simplify the expression 2(square root of 9) + 5(square root of 9), and both will lead you to the same answer of 21. Since both terms have the same square root, you can add them together first, and then simplify the square root of nine:

2(square root of 9) + 5(square root of 9)

= 7(square root of 9)

=7(3)

=21

Another way to simplify this expression is to simplify each perfect square first:

2(square root of 9) + 5(square root of 9)

=2(3) + 5(3)

=6 + 15

= 21