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SAT Mathematics Level 2: Help and Review22 chapters | 225 lessons

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.

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.

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.

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 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.

Let's try another one. To make it easier to follow, we've color coded the 'like' terms.

First let's combine the 'like' terms we already have:

Now let's simplify the terms that are already perfect squares (the 6√9 and the 3√4). We take the square root, and multiply it by the coefficient.

Now let's break up the 27 into 3 x 9 (which is a perfect square), and combine our two terms that don't have a radical.

Now we'll take the 9 out from under the radical by taking its square root,3, by its coefficient, 5

We have two like terms again, so we'll combine them as the last step.

And you're finished! Notice that no matter how complicated these problems get, just a few simple operations will reduce them to their simplest form. Although it's not necessary for accuracy, it's good form to rewrite these solutions in a logical order (such as by increasing size of the radical expression, as shown below).

A **square root** of any number of interest is the part of that number that may be squared to produce it. 3 x 3 = 9, so the square root of 9 is 3. 1 x 1 = 1, so the square root of 1 is 1. 2.5 x 2.5 = 6.25, so the square root of 6.25 is 2.5. Square roots may be added by converting them to their decimal values and then adding them, but the result is not exact. To add square roots (radical expressions) exactly, you may only reduce them and then add the 'like' terms (square roots with the same number under the radical, or √).

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SAT Mathematics Level 2: Help and Review22 chapters | 225 lessons

- Evaluating Square Roots of Perfect Squares 5:12
- Estimating Square Roots 5:10
- Simplifying Square Roots When not a Perfect Square 4:45
- Simplifying Expressions Containing Square Roots 7:03
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- Square Root: Sign, Rules & Problems 10:15
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