Video: Simplifying Radicals: Minimizing the Radicand

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).

11:22

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 this 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 be 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. Now, 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 25 plus nine, which is 34.

And taking square root of both sides, we’ve got 𝑥 is equal to the square root of 34. Now, that is an exact answer. And if you evaluated the square root of 34 on your calculator, you’d get 5.830951895, and so on, going on forever in fact. Well, we could round that to one decimal place or two decimal places. Now, that rounded number is obviously not 100 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 34.

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 root, 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 50.

Well, let’s think about the factors of 50. One times 50 are 50. Two times 25 are 50. Three doesn’t go, four doesn’t go, five times 10. And that’s it; there are all the factors. And looking at all of the factors, we can see that 25 is actually a square number. So, let’s rewrite the square root of 50 as the square root of 25 times two. And the square root of 25 times two can be rewritten as the square root of 25 times the square root of two. And as we said, 25 is a square number, so the square root of 25 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 50. Because the radicand, the contents of that square root sign there, are smaller than 50. So, five root two is simpler than root 50.

Now, let’s simplify the square root of 32.

Well, 32 has got six factors, one and 32, two and 16, and four and eight. And if we look carefully, we can see that two of those are square numbers, 16 and four. So, we’re gonna take the largest, 16. And that means we can rewrite root 32 as root 16 times two. And then, we could separate out the 16 and the two. So, we get root 16 times root two. And 16 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’ve simplified that expression. Now, if we’d have chosen the other square factor of 32, four, we could’ve rewritten root 32 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 32. So, we have simplified the expression root 32. 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’ve simplified the radical, root 32, 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 27.

So, first of all, let’s find out the factors of 27. And they are one and 27, and three and nine. Well, nine is a square factor, so the square root of 27 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 that 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 20, so six times root 20.

Let’s write down the factors of 20. And they are one and 20, two and 10, 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 20 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 12, so the answer is 12 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 18 plus the square root of 32.

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 as 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 18. Well, 18 has got six factors, one and 18, 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 18 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 18 becomes three root two. Again, that won’t simplify any further.

Lastly then, let’s think about the factors of 32. Well, 32 also got six factors, one and 32, two and 16, and four and eight. So, the largest square factor there is 16. So, root 32 is the root of 16 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 18 plus root 32 is root two plus two root two plus three root two plus four root two. And root two just means one lot of root two. So, that means we’ve got one of these root twos plus another two plus another three plus another four. That makes 10 in total. So, our answer is 10 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 than 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 yeah, four is a square number. Half of 18 is nine, and nine is a square number. And half of 32 is 16, so 16 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 given clues in the question. For example, if you were asked to fully simplify the square root of 80, 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, 80 divided by two is 40, and that’s not a square number. 80 doesn’t divide exactly by three. 80 divided by four is 20, and that’s not a square number. And 80 divided by five is 16. And that is a square number. So, 16 is gonna be our larger square factor.

So, root 80 is root 16 times five. And then, still, we don’t have to write out all the that equals root 16 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 we’d have had to write down.

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