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Squares and Square Roots

Squares and Square Roots


To multiply radicals, we multiply the radicands. Math can be straightforward, too. Who knew?

Sample Problem

This makes sense because the square root of 20 is a number that gives us 20 when multiplied by itself. Not convinced? We look incredibly trustworthy, so no clue why. We'll take you step by step and show you why it makes sense.

Multiplying by itself yields:

Therefore, is a square root of 20. Voila!

In general, if x and y are non-negative numbers (which means that we can take their square roots), then:

Fair enough. Unfortunately, this doesn't work with chromosomes. Sorry to rain on your "Eureka!" moment.

Sample Problems


A simplified radical term has only one radical sign. For example, the expression is simpler than .

Plus, isn't it nicer to look at? Not to sound shallow, but looks are everything when it comes to math. Well, almost everything. Accuracy is kind of important, too. Also, a nice personality.

We can also "un-multiply" something by breaking a radicand into factors. "Hey, isn't that dividing?" Shh. Maybe. If x and y are non-negative, then:

This gives us a way to simplify radical expressions by factoring the radicand. We wouldn't do it for fun, but we're all for anything that helps us get to the bottom of a particular problem.

Sample Problem

What's the square root of 20 in its most simplified form?

Since we can factor 20 as 4 × 5, we can rewrite the square root of 20 like this:

Since the square root of 4 is 2, we can re-rewrite it as:

We now have a radicand of 5 instead of a radicand of 20. Since none of the factors of 5 are perfect squares, we're done. That was a relatively painless root canal.

Sample Problem

Simplify .

We can factor 90 as 9 × 10. Therefore:

Note that we could also have factored 90 as 6 × 15, 5 × 18, 3 × 30, or 2 × 45. We chose the factorization 9 × 10 because 9 is a perfect square, so the radical goes away when we simplify . We want the radical to go away because it's mean to waiters and keeps talking about the time it saw Carrot Top in Vegas.

Factors that are perfect squares are ideal for factoring a radicand to simplify things, as we can see in the previous example. These are the factors that'll simplify to terms without a radical.

Sometimes to simplify we want to multiply radicals together, and sometimes we want to break them apart. It depends on how destructive a mood we're in. Fine, it really depends on what will be most helpful in solving a problem. Take out your aggression on your punching bag.

Sample Problem

Simplify .

Multiply the radicals together, then simplify:

If you think someone is super attractive (and also has a fantastic personality), tell your friends that he or she is a "perfect ." They probably won't have any idea what you're talking about, but that's fine. It keeps you interesting.

So far, all the terms we've been multiplying and un-multiplying have involved numbers only. You knew that had to be too good to last. Simplifying, multiplying, and breaking up radicals can also be done when we have variables in the radicand. Let's give x a little love and see how that's done.

Sample Problem

If x ≥ 0, then . We can multiply x by itself to get x2, so x is the square root of x2.

Sample Problem

If x < 0, then . You'll need to put on your thinking cap for this one. No, not that thinking cap; the one your Aunt Judy knitted for you this past winter, with the watermelon-shaped sequins. It'll make her happy.

We know will be positive, and the principal square root of x2 will therefore be positive. Since is positive and x is negative, it can't be true that .

However, -x is positive (think about the number line: if x is less than zero, than putting a negative sign in front of it will reflect it across zero and make it positive) and the square of -x is x2, so it is true that .

Did you follow all that? Don't hesitate to give it a second read to help cement it in your brain. Remember to put up a "Wet Cement" sign so someone walking by doesn't accidentally leave an impression on you.

To avoid complications like the previous example, some books simply assume that a variable that appears under a radical must be non-negative. We'll make this assumption too, but ask your teacher whether it's safe to make that assumption on your homework. He or she may have a different way of doing things. "But Shmoop said..." will rarely hold much water in class.

Sample Problem

Simplify .

Although it may not immediately look like it to the uninitiated, this is a perfect square. We have 4 copies of x as our radicand, or (x)(x)(x)(x). To find x4, we multiply x2 by itself.

Sample Problem

Simplify .

You might wish that we were still on numbers and think that this variable stuff is for the birds, but it's not so bad. We'll handle this the same way we would with numbers; we just need to keep in mind that the exponent is telling us how many copies we have of each variable.

We can break this up into:

And we can break up the first radical further as:

We can't simplify , but we can simplify the other two radicals:

Note that the y on the end is not in the radicand. It's only hanging out on the end there, sort of like a radicand groupie. Because someone might easily assume that it is part of the radicand, we usually put the radical at the end of a radical term. Re-order and write our final answer like so:

Not so bad. Are you regretting that you came down so hard on variables? Maybe you should've given them more of a chance before cutting them out of every photograph you were in together.

Just like we did with radicands that only had numbers, we still try to factor out perfect squares when radicands have variables...even when both numbers and variables are hanging out. Ooh, fraternizing with the enemy.

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