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

Instructor:
*Damien Howard*

Damien has a master's degree in physics and has taught physics lab to college students.

In this lesson, you'll learn how to subtract square roots. We'll explore different methods of square root subtraction to get either an exact or an approximate answer to the problem you're working on.

When working a math problem with exponents, you may already know that raising a number to an exponential power is the same as multiplying that number by itself as many times as the exponent indicates.

This makes sense for whole numbers, but what do you do when your exponent is a fraction? When this happens, what you actually have is a **root**.

In this lesson, we're going to be sticking with one type of root, square roots, and learning how to subtract them. Though we'll only be looking at square roots, everything you look at here will work for all types of roots, as long as the two roots you are subtracting have the same index.

A root consists of three parts; the **index** that was just mentioned, the **radical symbol**, and the **radicand**. Though, for square roots, it is standard practice to leave the index implied and not shown.

In order to subtract square roots, we need to pay attention to their radicands. If the radicands are the same, we can subtract the square roots by combining terms.

As you can see, the answer we get is in the form of a number multiplied by a square root. This is the simplest form this example subtraction problem can take as long as we're looking for an exact answer. We'll see later that a little more can be done if you're willing to accept an approximation instead.

Now, what do you do when your square roots have unlike radicands? First, you need to check and see if you can factor any of your radicands.

When factoring your radicands what you want to be on the lookout for are any perfect squares. A **perfect square** is any number created by multiplying another number by itself. In other words, it's the answer you get when you square another number. This means you can simplify your radicands by taking the square roots of any perfect square factors in them.

If your equation's square roots now have the same radicands, you can subtract just like we did in the previous section. If the radicands are still different, then you can't perform the subtraction. The exact answer for your equation is already at its simplest form.

So far in this lesson, we've only been working with whole numbers, the set of numbers including zero, and all positive integers. Things change a bit if we also allow decimals. With decimals you can use a calculator to get an approximation for a square root of non-perfect square numbers.

Working out square roots this way will allow you to solve the problem with your normal order of operations, just like you do any standard math problem. Even our problem that we said was in its simplest form earlier can actually be further approximated with decimals.

However, remember that working out square root subtraction with this method always gets an approximation, and not an exact answer. Often in math we want an exact answer, so you won't always be able to work out a problem this way. Learning your rules for subtraction of square roots from the previous sections is still important.

When you have an exponent in fractional form, what you actually have is a **root**. These roots have three parts; the **index**, the **radical symbol**, and the **radicand**. For square roots the index is normally implied.

When subtracting square roots with like radicands, you can subtract by combining terms.

If you don't have like radicands, you can see if the radicand can be further simplified by finding its **perfect-square** factors, and taking their square roots. If doing this gives you square roots in your equation with the same radicands, you can subtract by combining terms again. If you don't find like radicands, you cannot subtract to get an exact answer.

If the square root subtraction problem allows you to get an approximate answer instead of an exact one, you can use a calculator to find the approximate value for your square roots. You would then solve the problem like any other math problem by using normal order of operations.

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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
- Discriminant: Definition & Explanation
- Principal Square Root: Definition & Example 5:02
- Square Root: Sign, Rules & Problems 10:15
- The Square Root Property 6:49
- Adding Square Roots
- Subtracting Square Roots
- Go to Square Roots: Help and Review

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