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A radical term is any term that has at least one radical in it. Radical terms can be messy, so we want to write them in the simplest way possible. It's probably also a good idea to fit them with a bib to catch any wayward mashed banana.
Like karate, we only use radicals when we need to. If a radicand is a perfect square, we scrap both the radicand and its radical and write the square root in their place. This is the same thing we did in the previous section when we found square roots, but now we're calling it "simplifying." We've got like a million more names for it, so hit us up any time. We're a river to our people.
To simplify the radical term if a radicand is not a perfect square, we write an equivalent expression in which
We use multiplication and division to perform this simplification. That's the end of this section. We'll celebrate by starting a new one.
To multiply radicals, we multiply the radicands. Math can be straightforward, too. Who knew?
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.
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.
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.
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.
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.
If x ≥ 0, then . We can multiply x by itself to get x2, so x is the square root of x2.
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.
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.
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.
These square roots are nice and everything, but it would be nicer if we could turn them into fractions on top of it. It's like we can read your mind, right? Spooky.
To divide radicals, we divide the radicands.
Here's why this makes sense: if we multiply by itself, we'll find . As a result, is a square root of . You may have already known that. It was probably on your "division of radicals" wallpaper when you were growing up.
Same deal with variables. If x and y are any non-negative numbers (so that we can take their square roots), then:
Divide, simplifying if possible:
The rule for division with radicals can also be used in reverse. Check your rearview mirror before backing up first. You don't want to run over any variables. Or do you?
The rule can be used to break up an expression with a fractional radicand into a quotient of radicals with nicer radicands. If x and y are non-negative integers, then:
These are both numbers from our list, so we can tell right away that this won't be too painful. We need to be able to deal with each of these numbers separately, so we rewrite this expression as a quotient of radicals and simplify from there.
Sometimes it's more helpful to rewrite the square root of a fraction as a fraction of square roots, as in the previous example. Sometimes, however, it's more efficient to do the division in the radicand. We wish we could tell you it was more cut-and-dried than that, but you'll need to use your own judgment. Wear a black robe and powdered wig if you think it'll help.
Turning this into a quotient of radicals yields .
Since 28 = 4 × 7, let's rewrite the whole thing:
If we had done the division in the radicand first, however, we would have gotten:
Well, that was way easier. We're on board with this method; how about you?
Which method do we use when? In general, if a radicand is a fraction that can be simplified to a perfect square or some other nice, even number (for example, an integer times a perfect square), simplify the radicand first. If the radicand is a fraction that doesn't simplify to anything we like, it's probably time to split up the expression into a quotient of square roots. Either way, simplifying correctly will give us the correct answer. If we wind up going the long way around every once in a while, so be it. At least we can admire the scenery along the way.
You might have noticed that none of the division problems so far have had answers where a radical showed up in the denominator. We had a problem with the answer , but nothing with an answer like . This is because it works a little differently, and we haven't explained to you yet what to do with that sort of radical term. We will now, though. Hallelujah, the wait is over!
When it comes to writing fractions with radicals in them, you'll find that most teachers don't like to see radicals in the denominator. They would look at and be unhappy. No one wants your teacher to be unhappy, least of all your teacher. To keep your teacher happy, multiply your fraction by a clever form of 1:
Your teacher might be so thrilled that she'll decide to give everyone A's for the quarter. It might be a combination of your actions or the Lord of the Rings extended edition marathon she sat through last weekend, but either way you'll take it.
The expressions are mathematically equivalent. However, since radicals can be yucky (that's the technical term for it) and having radicals in denominators can be even more yucky, most people prefer the second expression.
Here's another way to see that and are equivalent, in case you're a naturally skeptical person and need more convincing. Since , we can rewrite like this:
Then we can cancel a factor of from both the numerator and denominator:
That should satisfy even the most adamant of skeptics.
Rewrite the expression without a radical in the denominator.
First we whip out our clever form of 1. Multiplying by gets us to .
The process of rewriting so that we don't have a radical in the denominator is called rationalizing the denominator, because we're writing the denominator as a rational number instead of as some weird square root. That makes you, your teacher, Shmoop, and mathematicians everywhere happy. Everyone wins!
Before rationalizing the denominator, it's helpful to simplify the radical in the denominator, if possible.
Simplify the radical term .
Way 1: First simplify the radical in the denominator, then rationalize. First we factor 28:
28 = 4 × 7
And then we can break up as:
This lets us simplify the original expression a lil bit:
From here, we can rationalize the denominator:
Way 2: This time, we rationalize the denominator before simplifying. To rationalize the denominator, we multiply by a clever form of 1.
Now we need to simplify. Since 28 = 4 × 7, we know , and so:
Thankfully, we got the same answer with both Way 1 and Way 2. There's also Way 3: sneak a peek down the page and steal our answer after we do all the hard work, but that won't help you much in a test-taking situation. Also, it's not as personally fulfilling.
The moral of the story is that we can rationalize the denominator first or simplify the radical first, as long as we do both before writing down our final answer. It may be more efficient to simplify the radical in the denominator first, but it's certainly not the end of the world if you don't. It's a good thing, too, because that would be far too much power for a single individual.
Of course, we can do the same sort of thing when there are variables in the radicands. Oh, you knew we were going there eventually.
This seems fairly straightforward. We break up the square root into a quotient of two square roots:
It looks like we're done, but we need to eliminate that pesky radical in the denominator first. Simply shooing it away and threatening it with a radical-swatter doesn't appear to be working. To accomplish our goal, we rationalize the denominator.
In other words, we multiply by a clever form of 1: