Subtracting Square Roots

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.

Square Roots

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.

powers as multiplication

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.

roots as powers

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.

Subtracting with Like Radicands

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.

root parts

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.

square root subtraction 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.

Subtracting with Unlike Radicands

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.

perfect squares in square roots

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.

square root subtraction through factoring of radicals

example of square roots that can not be subtracted


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.

square root approximation

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.

square root subtraction through approximation

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