The word radical has a lot of interesting definitions, but radical arithmetic doesn't actually refer to arithmetic that favors drastic political, economic, or social reforms. We're talking about doing arithmetic with expressions that contain radical symbols. Sorry, you can put those signs down.
We've seen two special types of expressions so far: polynomial expressions, and rational expressions. Now we'll add one more special type, because things are funnier in threes: a radical expression is any expression with one or more radical signs in it. Another way to put this is that a radical expression has at least one radical term, or a term with at least one radical in it. Yet one more way to put this, because things are funnier in threes, is that a radical expression radiates with radicalocity.
Hmm, not so much. So much for the "rule of threes."
To add or subtract radical expressions, simplify each radical term and then combine like terms. A simplified radical term consists of a coefficient and a radical, under which there is a radicand. Can you believe you understand now what all these crazy words mean? It's like you can speak a secret language.
In the term , the coefficient is 5 and the radicand is 7.
In the term , we need to rationalize the denominator to find .
To put this in the form we need, it could be rewritten as:
So the coefficient is and the radicand is 35. We still haven't gotten rid of the fraction line, but at least it isn't combined with the square root symbol any longer. That many weird symbols consorting together makes us nervous. It feels like they're up to something.
Two radical terms are considered like terms if they have the same radicand. This makes them "term twins." You'll be able to tell, because they're always finishing each other's sentences.
Be careful: It's important to simplify radical terms before combining like terms. Sometimes two terms can be rewritten as like terms, but we can't see it until we simplify. It's the same way you can't switch one kid out for another without their parents noticing until you've first made sure that they're twins. Otherwise, it's called kidnapping.
Radical expressions may contain variables either outside or inside the radicands. After simplifying, we can combine like terms in the same way we did when only numbers were involved. Except now it's more fun, because we can use variables!
...we'll keep telling ourselves that until it feels true.
Now that we know how to figure out which terms can be combined, we'll combine some. How about that.
This is similar to adding or subtracting variables. In the same way that 3x + 4x = 7x, so does .
To add or subtract like radical terms, we add or subtract the coefficients. We don't do anything to the radicands, which is why we made sure they were the same in the first place.
We keep the radicand the same, and add the coefficients 3 and 8:
First, simplify each radical. Then we can rewrite the problem as:
We can simplify the first radical term and rewrite the problem as:
We can't combine these terms since the radicands aren't the same, so that's our final answer.
We can also do this sort of thing with expressions that have variables. Variables and numbers have sort of an "anything you can do, I can do better" relationship, or at least an "anything you can do, I can do equally" one.
Multiplication of radical expressions is similar to multiplication of polynomials. Remember what a gas that was? Now we can experience that thrill ride all over again, and you don't even need to wait in an incredibly long line first.
When multiplying radical expressions, we give the answer in simplified form. Multiplying two monomial (one-term) radical expressions is the same thing as simplifying a radical term.
We multiply the radicands to find .
Then, we simplify our answer to .
We distribute the and simplify the resulting terms:
Since these simplified terms have different radicands, there are no like terms to combine, so we're done. If you absolutely need a combining fix, we suggest experimenting with your little sister's poster paints. Hint: yellow and blue make green.
To find the product of two binomial (two-term) radical expressions, we use the distributive property. Remember him?
The first thing we do is simplify each radical term, if possible. We can replace with 2, and now the problem is:
Now it's distribution time. We're gonna multiply the first term in the first set of parentheses by both terms in the second set, then multiply the -2x by both terms in the second set. It's all coming back to you now, right? Don't give us that look, everyone loves Celine.
Anyway, we start by multiplying the first terms:
That gives us:
Then we multiply that first term by the second term in the second set of parentheses:
...which gives us:
Now we distribute the -2x to both terms in the second set, starting with the square root of 3:
That gets us:
Finally, we multiply the last terms:
...which gives us our very last term:
Add together all of our cute little products to get our final answer:
Since the radicals have different radicands—one might even say "radically different radicands," which we will—this is as simplified as the answer gets. Rad.
We already know how to divide one radical term by another, and how to simplify our answer by rationalizing denominators. However, because you looked sleepy the last time we went over it, let's review.
Since 5 divides nicely into 25, we can divide the radicands:
There are two other kinds of division with radical expressions that you're likely to be asked about by a teacher, or on a test, or even by a curious stranger on the street: dividing a multinomial by a single term, and dividing a multinomial or binomial by a binomial. Adjust your goggles and take a deep breath, because we're diving in.
Dividing a multinomial by a single term is similar to dividing a polynomial by a single term. We break up the quotient into several simpler quotients, simplify those, and add them together. This isn't our own idea; we're just trying to maintain the status quotient. Ba dum tsch!
Break up the quotient into several nicer quotients. If you're using a hammer, watch your thumbs.
We can simplify each of these quotients:
Adding the simplified forms gives us:
Now we're almost done. Hey, the first and last terms have the same radicand, so we can group 'em together:
Now we're done. We promise. Pinky swear. No, seriously, put your pinky up to the monitor. Then, wait until we come out of the computer to meet you.
Dividing by a binomial is a little weirder. Not like Lady Gaga weird, though; maybe only Tom Cruise weird. We'll show how this works with an example.
Multiply by a clever form of 1. We just learned this, and this is crazy, but trust us. It will work, and we'll explain why we chose this crazy-looking clever form of 1 after the example. Hang in there! And call us maybe? On second though, don't. You have algebra to learn.
Now we multiply things out. Hold on, things will get scary for a second.
Notice what happened in the denominator: we multiplied two binomials of the form .
After multiplying, the radical went away because we had a difference of two squares.
Radical expressions of the form are called conjugates. Go ahead and file that away for future reference. Preferably under "C," so you'll be able to find it when you need it.
When we have a quotient of expressions where the denominator has two terms and contains at least one radical, we can rationalize the denominator, or eliminate the radical(s) in the denominator. (We think "rationalizing" something sounds better than "eliminating" it. What can we say, we're softies like that.) We rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator.
Or, in other words: put that sign down, flip it, and reverse it.