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Video: Simplifying Radicals: Minimizing the Radicand

Tim Burnham

We explain the terms radical, radicand, and surd, then work through a series of examples to demonstrate how to factor out square factors from radicands in order to simplify radical expressions (minimizing the value under the root).


Video Transcript

In this video, we’re gonna look at one of the ways of simplifying radicals. You may know this topic as simplify surds. We’re gonna be looking at how to manipulate the expression so that the radicand, that’s the number under the radical or the root sign, is as small as possible. Then we’ll go on to solve some typical questions which require you to use the skill.

First let’s remind ourselves of some terminology. This symbol here, looks like a big tick, is called the radical, or some people call it the root sign. This number here is called the index, and that tells you which root you’re looking for. So I’ll been looking for the square root, the cube root, the fourth root, or whatever. And the number that you’re taking the radical or root of is called the radicand. And when the index is two, we often don’t bother writing it. So instead of writing square root of five like this, we tend to write it like this.

Now let’s just quickly think about why we keep numbers in this format, why we work with expressions like this. Well for example, we could have a right triangle like this with one side has got a length of three, one side has got a length of five, and we want to know the length of the other side. Well, we can use the Pythagorean theorem; and that states for in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other sides. So in our triangle, 𝑥 squared is equal to five squared plus three squared, which is twenty-five plus nine, which is thirty-four. And taking square root of both sides, we’ve got 𝑥 is equal to the square root of thirty-four. Now that is an exact answer. And if you evaluated the square root of thirty-four on your calculator you get five point eight three o nine five one eight nine five, and so on, going on forever in fact. Well we can round that to one decimal place or two decimal places. Now that rounded number is obviously not a hundred percent accurate. But if you were building a wooden frame to those dimensions for example, there’s a limit to how accurately you could cut the wood. So for practical purposes, one or two decimal places will probably be a very convenient way to present the answer to a carpenter who’s making the frame. But in the pure world of mathematics, we’d rather work with our completely accurate answer, the square root of thirty-four.

Now a set of conventions has evolved around radicals, about how we present them. And one of these is to try to keep the number under the roof the radicand as small as possible. So let’s take a look at some situations where we can simplify the radicals using this convention.

Simplify the square root of fifty. Well let’s think about the factors of fifty: one times fifty of fifty, two times twenty-five of fifty, three doesn’t go, four doesn’t go, five times ten. And that’s it; there are all the factors. And looking all of the factors, we can see the twenty-five is actually a square number. So let’s rewrite the square root of fifty as the square root of twenty-five times two. And the square root of twenty-five times two can be rewritten as the square root of twenty-five times the square root of two. And as we said, twenty-five is a square number, so the square root of twenty-five is exactly five. So that becomes five times the square root of two, and of course we don’t need to put the multiplication sign between the two of them. We can just express it as five root two. So according to that convention, five root two is considered to be simpler than root fifty because the radicand the contents of that square root sign there are smaller than fifty so five root two is simpler than root fifty.

Now let’s simplify the square root of thirty-two. Well thirty-two has got six factors: one and thirty-two, two and sixteen, and four and eight. And if we look carefully, we can see the two of those are square numbers: sixteen and four. So we’re gonna take the largest, sixteen. And that means we can rewrite root thirty-two as root sixteen times two, and then we can separate out the sixteen and the two. So we get root sixteen times root two. And sixteen being a square number means that the square root is exactly four. So that becomes four times root two, or as we’d normally write it: four root two. And we can see that the radicand is smaller than it was before, so we simplified that expression.

Now if we’d have chosen the other square factor of thirty-two, four, we could have rewritten root thirty-two as root four times eight, which is root four times root eight, which is two root eight. And again that radicand is smaller than it was before; eight is smaller than thirty-two. So we have simplified the expression root thirty-two. But that eight has a square factor, four times two is eight. And four, as we said before, is a square number. So although we simplified the radical, root thirty-two, we haven’t fully simplified it because what we’ve got left in this radical here could still be simplified further. So it’s equal to two times root four times two, and we can split up the four and the two into their own roots, and then root four is two. So that becomes two times two times root two, which is equal to four root two, which is the answer that we got before. So if you want to fully simplify a radical, make sure that you get into the habit of always checking the radicand in your answer to see if it’s got any square factors. If it has got square factors, then you can simplify it down further.

Okay, I want you to pause the video and to have a go at this yourself, so I’ll just wait three seconds and then I’ll go through the answer. Well, we’ve got to fully simplify the square root of twenty-seven. So first of all, let’s find out the factors of twenty-seven. And they are one and twenty-seven and three and nine. Well nine is a square factor, so the square root of twenty-seven is root nine times three. And of course that splits up into root nine times root three. The square root of nine is three. So our answer is three root three. And just checking the radicand three, the factors of three are only one and three. None of those are square numbers, so we can’t simplify any further.

Now we’ve got to fully simplify six root twenty, so six times root twenty. Let’s write down the factors of twenty. And they are one and twenty, two and ten, four and five. Well the biggest square factor there is four. In fact, the only square factor is four, so we can rewrite six root twenty as six times the square root of four times five. And that’s the same as six times the square root of four times the square root of five. And the square root of four of course is two, so that becomes six times two times the square root of five. And six times two is twelve, so the answer is twelve root five. And again looking at the radicand five, that’s got no square factors so we’ve simplified this as fully as we can.

Lastly then, let’s look at this question: fully simplify the square root of two plus the square root of eight plus the square root of eighteen plus the square root of thirty-two. So we’re just gonna go through each of these terms individually and see if we can simplify them down. Well root two, that radicand doesn’t have any square factors so that’s a simple as it can go. So let’s consider root eight. And eight has got the factors one and eight and two and four. And four is a square number, so we can rewrite root eight as the square root of four times two, which is the square root of four times the square root of two. And the square root of four is two, So root eight is the same as two root two. And root two, that doesn’t have any square factors, so that’s fully simplified. Next let’s consider root eighteen. Well, eighteens got six factors: one and eighteen, two and nine, and three and six. And nine is the largest square factor, that is the only square Factor, so we’re gonna be able to simplify this one. root eighteen is equal to the root of nine times two, which can be split into the root of nine times the root of two. And the square root of nine is three, so root eighteen becomes three root two. Again, that won’t simplify any further. Lastly then, let’s think about the factors of thirty-two. Well thirty-twos also got six factors: one and thirty-two, two and sixteen, and four and eight. So the largest square factor there is sixteen. So root thirty-two is the root of sixteen times two. So that simplifies to four root two. Now, we can start trying to simplify our expression. So root two plus root eight plus root eighteen plus root thirty-two is root two plus two root two plus three root two plus four root two. And root two just means one root two. So that means we’ve got one of these root twos plus another two plus another three plus another four; that makes ten in total. So our answer is ten root two.

Now that’s quite a lot of working out for a pretty short little question, so let’s talk about a couple of little strategies that mainly you can do a lot of this in your head rather have to write quite so much working out down. Now, the first strategy is a bit of exam technique. If you’ve been given this sort of question in a test or an exam, you’ve been told that the first term is root two; now the chances are that these others are gonna be multiples of root two as well. And that should guide you for starting points for trying to work out the simplified versions of each of these other terms. Now if these are all gonna be multiples of root two, then we’re gonna be able to factor those things down into something times two. And hopefully, they’re going to be square numbers. So let’s just try that. Half of eight is four. And yes, four is a square number. Half of eighteen is nine, and nine is a square number. And half of thirty-two is sixteen, so sixteen is a square number. So we didn’t have to work out all the factors of those numbers; the question gave us a clue for a good starting point to start making guesses as to what those numbers might be. And you can apply some techniques even if you’re not giving clues in the question.

For example, if you were asked to fully simplify the square root of eighty, you could try dividing by two then three then four then five. And as soon as you get a square factor, you know you got the largest square factor. So eighty divided by two is forty, and that’s not a square number; eighty doesn’t divide exactly by three; eighty divided by four is twenty, and that’s not a square number; and eighty divided by five is sixteen, and that is a square number. So sixteen is gonna be our larger square factor. So root eighty is root sixteen times five. And then, still, we don’t have to write out all that equals root sixteen times root five, and that equals four times root five. We can jump straight to the answer cause we can do that bit in our head. So all of this stuff here would probably have been done in our head, and this is the only working out would’ve had to write down.