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Video: Simplifying Radicals - Rationalising the Denominator

Tim Burnham

Simplifying a radical expression, or surd, by multiplying its numerator and denominator by a suitable radical expression to rationalize its denominator.

13:22

Video Transcript

Let’s have a look at simplifying some radicals. Now depending where you live, you might use terms like irrational numbers or surds. But the specific aspect of this that we’re going to be looking at today is rationalising denominators. So if we’ve got expressions like this one one over the square root of two the denominator, the “downstairs”, of that fraction has got a radical or it’s got a surd number just sitting there: the square root of two. Now that looks like quite a simple number to most of us. But within mathematical circles, we don’t like that situation where you’ve got those radicals on the denominator, so we have to try and eliminate those. And that’s really what we gonna be looking at in this video.

So in our first example, rationalise the denominator of one over the square root of five. Okay so let’s just write out one over root five equals one over root five. Well that seems pretty obvious; it’s equal to itself. But let’s try this then, so one over root five times one. If we times anything by one then you’re gonna get the same value. So that’s clearly true as well. Now we’re gonna look at different flavours of one different versions of one.

Okay, so root five over root five, something divided by itself will give you the answer one, so clearly root five over root five is one. So we’ve done the same thing here; that is still one over root five times one. Now the reason I picked root five over root five is because look the denominator here contains root five; I know that root five times root five is just five. And five is not a radical number; it’s not a surd; it’s not in root format; it’s a- it’s a rational number. So on the numerator, one times root five is just root five, and root five times root five is five.

So there’s our answer. Now, you know this is supposed to be simplifying these radicals; and in fact, this-this answer here and probably looks a little bit more complicated than this answer here. This has is got smaller digits in it, smaller and numbers on the numerator and the denominator. But nonetheless, to simplify a surd or to simplify a radical in this format, we want to eliminate these radicals from the denominator. So that’s the answer that we’re looking for.

So let’s look at another example then: four over root twelve. So the basic process is gonna be just what we did before, but there might be a little bit more cancelling down to be done at the end as well. So as before, we said four over root twelve is equal to four over root twelve times one, and the version of one that we’re going to be using is — yep, you guessed it. It’s gonna be root twelve over root twelve so that we can eliminate the radical from the denominator.

So there we have that, root twelve over root twelve is just one, so all we’ve done is we’ve taken the fraction that we were given and we multiplied it by one. So we’re not changing the magnitude, the size, of this fraction. So multiplying those two fractions together, we’ve got one big fraction four times root twelve on the top and root twelve times root twelve on the bottom. So remember, root twelve times root twelve is just twelve.

And the numerator is still four root twelve four, four times root twelve. So we’ve got four times root twelve on the numerator and twelve on the denominator. Well look four and twelve, they’ve got a common factor of four. So four divided by four is just one; twelve divided by four is three. So I’ve cancelled that down a little bit and simplified it.

So we can write that out as root twelve over three. Now actually, that’s not our answer because given that we’re trying to — we have rationalised the denominator, so technically we have answered that question. But typically this would be simplify- fully simplified this radical; and in that case, we’ve got a way of simplifying the numerator a little bit further as well. Now the temptation is to cancel the three with the twelve. But remember because it’s root twelve on the top, you can’t do that because it’s not root three on the bottom, so we can’t do any cancelling there. But twelve can be written as four times three, and four is a square number so it’s a square factor of twelve.

And root of four times three is the same as the square root of four times the square root of three, and of course the square root of four is two. So that gives us two root three over three. So we’ve just illustrated there is that the- when we’re simplifying these surds, these radicals, we always looking to remove radicals from the denominator. But we’re also looking to make the contents of these square roots, the number under the root if you like, as small as it possibly can be. And by factorising out this square factor of twelve, four and then evaluating that the square root of four is two, we’ve managed to make this number, this value, simpler as we would say in mathematical terms.

Well, sometimes the fractions are a little trickier. So in this question we’ve been asked to fully simplify fifteen plus root three, which is on the numerator, over root three. Now world’s number one mistake in terms of these things is just cancelling off the root threes and getting the question wrong. But you can’t do that; remember, you can only cancel factors. And we’ve got fifteen plus root three on the top, not fifteen times root three, so we can’t cancel down those factors like that. Okay, let’s look at what we can do. We’ve got a root three on the denominator. So if I multiply that fraction by one, but the version of one I’m gonna use is root three over root three.

So I bracketed it together that fifteen plus root three just to make sure that we know that they’re together. And I’m gonna multiply the whole of that numerator by root three and I’m gonna multiply the whole of the denominator by root three, remembering that root three divided by root three is one, so we’re just multiplying by one. Okay, so the reason we’ve done that is because when we multiply these two things together, root three times root three, it just gives us three. That’s gonna eliminate our radical from the denominator.

So now all I’ve got to do is multiply out the terms of the numerator, so root three times root three is three, and root three times fifteen is fifteen root three. Now again we’re gonna look to see: can we factorize? Can we cancel? What can we do? Now if we look at that numerator there, I’ve got fifteen root three plus three. Well three is a factor of three and three is also a factor of fifteen. So I could simplify that numerator a bit by factorising it.

and three lots of five root three is fifteen root three and three lots of one make three over here. So I’ve just factorised the numerator there. Now it’s probably worth mentioning now that I’ve got three times the whole of five root three plus one and I’ve got three on the bottom. So now we can cancel things out, so three is a factor of three. If I divide three by three, I get one. If I divide three by three I get one. So I’ve got one lot of five root three plus one all over one. Well clearly.

I don’t need to have the multiply one and divide by one in there. I can just simplify that to five root three plus one or one plus five root three. Doesn’t matter which way you write those two round.

Okay, let’s look at this example then. Fully simplify one over root two plus one. So this time we’ve got a slightly more complicated term on the denominator rather than the numerator, and this does make life a little bit more difficult. Our general approach is just the same we’re going to multiply that by some version of one. But in this case the version of one that we gonna multiply by is root two minus one. So what you do for these ones, you have a look at the denominator. So we’ve got root two plus one and you just change the sign, and that gives us the term that we’re going to use to multiply those out. You’ll see why that is in a moment when we actually do the multiplication. For those of you are thinking ahead a little bit, think difference of two squares and what happens when you use the difference of two squares if you know about that. Okay, so when we multiply out the numerator I just got one lot of that whole bracket so that’s gonna to be fairly straightforward.

So that pa- next one, one lot of root two is positive root two and one lot of negative one is just negative one. So looking at our denominator, we’ve got the square root of two times the square root of two, which is just two. We’ve then got negative root two add root two. So if we start off at negative root two and we add it to itself, that’s just gonna give us zero. So those two things cancel out, so we’ve got two take away one.

And two take away one is just one, so we’ve got negative root- sorry! we’ve got root two take away one all over one. Well we don’t need to write over one. So the whole thing just simplifies down to the square root of two take away one.

Right then, one more example, so fully simplify nine over three minus the square root of three. Now it’s always a good idea to put denominators in brackets; if you got a couple of terms, then bracket them together, same again with the numerators so it’s clear which terms have to stick together. And we’ve got to find a version of one to multiply this by which is gonna completely get rid of all those radicals from the denominator in this fraction. Now what we said before is we look at it. We’ve got the term here; we’ve got three; we’ve got negative root three here. So we’re gonna — if we change that sign, it would give us three plus root three. That’s the term which we’re gonna put on the numerator and the denominator to make our version of one.

So in order to simplify this, what we’re gonna do is a nice complicated process where we multiply the top and the bottom of that fraction by three plus root three. So I’m not actually gonna multiply out the numerator just yet because there’s a chance, you know, with these questions that maybe if we factorise something later on, something might cancel out. So sort of save us a little bit of work, I’m not really gonna do anything for now with the numerator. But let’s look at the denominator, I’ve gotta do three times three, which gives us nine. I’m gonna do three times positive root three so they’re both positive, so it’s gonna be positive answer so positive three root three.

Then moving on to the next terms, we’ve got negative root three times three, so that’s negative three root three, and then negative three root three times positive root three. Well negative times positive’s gonna give us a negative, and root three times root three is three. That’s the definition of square roots.

So let’s see if we can tidy up that denominator a little bit. I’ve got nine and I’m taking away three, so that’s just six. And I’ve got positive three root three and then I’m taking away three root three fr- I’m taking my three root three from itself. So they’re gonna cancel out and give us zero, three root three take away three root three is zero. So there’s nothing else to put on that denominator.

Now I’ve got nine times three plus root three on the top and I’ve got six on the bottom. So because I’ve got things multiplied together, factors, I can do some cancelling. three is a factor of nine and of six, so six divided by three would be two, nine divided by three will be three. So that gives us our answer three lots of three plus root three all over two. Not really a lot of cancelling I can do there now; that is in fact my answer.

Now if I multiplied out that numerator and got nine plus three root three all over two, that is just as good an answer; that’s not a problem. And in fact, I could even split that out into two separate fractions nine over two plus three root three over two. All equivalent answers and all of them are correct. So when we talk about rationalising denominators, it just means getting rid of all these kind of square rooty type things from the denominator, and the way that we do it, remember, is multiply by some version of one, which is gonna help us to wipe out those Radicals from the denominator.