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Adding Square Roots

Instructor: Michael Quist

Michael has taught college-level mathematics and sociology; high school math, history, science, and speech/drama; and has a doctorate in education.

Square roots and radical expressions can be really difficult to add because they're often considered unlike terms and don't mix well. In this lesson, we'll look at how to simplify and solve problems involving adding square roots.

What is a Square Root?

Working with square roots and other radical expressions can be really challenging, but once you understand the principles, they can get a lot easier. You have to find 'like' expressions and then combine them according to a certain set of rules. In this lesson, we'll walk through it together.

A square root of a number is that part of the number that can be multiplied by itself to get back to the original number. For example, if you take 2 x 2, you get 4. 4 is the square of 2, which makes 2 the square root of 4. Some square roots are easy and obvious, like 2 and 4, while others are more complicated. For example, the number that is multiplied by itself to get to 2 is called an ''irrational'' number, which means there's no neat way to write it, except for its radical form, which is √2.

Adding the Decimal Form

One way to add square roots is to reduce them to their decimal form. This is especially easy if you have a calculator. For example, the square root of 2 is about 1.414, which means if you multiply 1.414 by itself you'll almost get back to 2 (about 1.999). If you want to add √2 (about 1.414) to √3 (about 1.732), you'd get about 3.146, which is approximately the sum of the two square roots.

Larger sequences of square roots can be added the same way:

3√40 + 5√10 + 7√3

= (approximately) 3(6.325) + 5(3.162) + 7(1.732)

= 18.974 + 15.811 + 12.124

= 46.909

Unfortunately, this is not an exact answer, and many math problems require an exact answer, even if you have to leave it in radical form. Here's how you get exact answers that include radicals.

Simplifying Radicals and 'Like' Terms

Adding the radical form of square roots is a lot like adding variable expressions. You have to get to a 'like' form before you can simplify them. Square roots are 'like' terms if they have the same value under the radical. For example, √2 and another √2 are 'like' terms, while √2 and √3 are not 'like' terms.

So can you simplify radicals to get to like terms? Yes, sometimes you can. For example, the square root of 8 may be rewritten as the square root of 4 times the square root of 2. √8 = √4 x √2. Since the square root of 4 is 2 (2 x 2 = 4), that means that √8 = 2 √2. This lets us add some square root terms that otherwise we would not be able to.

What about larger numbers? For example, what would you do with √200? 200 has no simple root, but it can be broken up into products that do have simple roots.

200 = 2 x 100, so √200 = √2 x √100 = 10√2.

108 = 3 x 36, so √108 = √3 x √36 = 6√3.

Adding Square Root Expressions in Radical Form

Adding radical forms of square roots is sort of like just stacking them up. For example, if I have 5 √2 to add to 7 √2, that means I really have 12 of them. Just add the numbers in front of the radicals (√). It's like the √2 guys are just packages, and you're counting how many of them you have. It's easy, if they're all like terms.

5√2 + 3√2 + √2 + 4√2 = (5 + 3 + 1 + 4)√2 = 13√ 2

You can simplify your terms to obtain 'like' terms. For example:

5√8 + 3√4 + √2 + 4√16

= 5(2(√2)) + 3(2) + √2 + 4(4)

= 10√2 + 6 + √2 + 16

= 22 + 11√2

Notice that the 6 and the 16 are 'like' terms, while the 10√2 and the √2 are also like terms. Once we've collected the 'like' terms, we're done.

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