Video: Multiplying Radical Expressions

We simplify radical terms by factoring out square factors from the radicands, then multiply the resultant terms without the use of a calculator. We multiply individual radical terms and more complicated expressions with multiple parentheses.

17:14

Video Transcript

In this video, we’re gonna be multiplying radical expressions and simplifying the results. Radical expressions, remember, are exact forms of numbers that involve radicals or root signs, like root five or seven root three and stuff like that.

Our first question then is, express five root five times root 45 in its simplest form.

Now, radicals in their simplest form have the radicands, these bits here under the root signs, as small as they can be after factoring out any square factors. So firstly, that five right up against the root five means five times root five. And next, we’ve got to simplify root 45. And now, we’re going to try and find the biggest square factor we can find of 45. And the basic technique for doing this is try dividing by two, then by three, then by four, then by five, and so on. And see if any of the results that we get are square numbers. And the first one we encounter is also gonna be the biggest square factor. So 45 divided by two doesn’t work. 45 divided by three is 15. But that’s not a square number. 45 divided by four doesn’t go into 45 exactly. 45 divided by five is nine. And that is a square number. So that is our largest square factor. So I’m gonna rewrite 45 as nine times five.

And now, root nine times five, we’re gonna split out into root nine times root five. And the reason for doing that is that root nine is three. Nine is a perfect square so root nine is just three. Now with multiplication, it doesn’t matter what order you multiply things together in. You’re gonna get the same result. So I’m gonna swap around the root five and three in the middle and multiply things together in this order. So we’ve got five times three is 15. And the square root of five times the square root of five is just five. And 15 times five is 75, so the answer is 75.

The next question then is express four root seven times two root 49 in its simplest form.

Now, a couple of things when we look at this, the four right up against the root seven means four times root seven. And the two right up against the root 49 means two times root 49. But also 49 is a perfect square, so root 49 is exactly seven. So we can rewrite this as four times root seven times two times seven. And again, with multiplication it doesn’t matter what order you multiply things together in. You’re gonna get the same answer. So I’m gonna to reorder those to four times two times seven times root seven. And four times two is eight and eight times seven is 56. So that’s 56 root seven. And if we look at seven, it’s — in terms of its factors, only one and seven are factors. So obviously one is a square number. But if we factor that one, it’s not gonna enable us to make that radicand any smaller. So 56 root seven is our answer.

And the next question is, use the distributive property of multiplication to expand five times three minus two times the square root of five.

Now, the distributive property of multiplication tells us that this means five times three take away five times two root five. And five times three is 15. And five times two root five, well that means five times two times root five. And five times two is 10, so that’s 10 root five. So we’ve got 15 minus 10 root five. And that won’t simplify down any further, so that is our answer.

Now, the next question is, simplify three root three times two plus five root three.

So we’re gonna use the distributive property of multiplication to do three root three times two. And then we’re going to add three root three times five root three. Now, let’s rewrite that with all the proper multiplication signs in there. And let’s swap things around a little bit because it doesn’t matter what order you multiply things together in. So that gives us three times two times root three plus three times five times root three times root three. Well, three times two is six. So this bit becomes six times root three or just six root three. And three times five is 15. And the square root of three times the square root of three is just three. So this gives us six root three plus 15 times three. Well, 15 times three is 45. So it’s important to take note of the order of operations. But this all simplifies down to six root three plus 45.

Now, our next question is littered with negative signs. So we’ve gotta be really, really careful.

We’ve got to simplify negative three root two times negative four take away two root two.

First of all, I’m gonna put parentheses around the negative numbers. Just to make sure that we know that they’re negative, and we don’t forget the negative sign. And then, we’re gonna use the distributive property of multiplication to do negative three root two times negative four. And then we’re gonna take away negative three root two times two root two. Now, this bit here just means negative three times root two times negative four. And because they’re multiplied together, I can put them in any order I like. And likewise, with these terms here, I’m going to put all the radical terms together and all the rest of the terms together.

Right, let’s break down these things and multiply things together. So I’ve got negative three times negative four, which is positive 12. And that is multiplied by root two. Then we’ve got negative three times two, which is negative six. And the square root of two times the square root of two, which is just two. So this overall right-hand expression here becomes negative six times two, which is negative 12. So we’ve ended up with 12 root two take away negative 12. Well, if we take away negative 12, that’s the same as adding 12. And root two won’t simplify down anymore, so this is our answer, 12 root two plus 12.

Next stop then, we’ve gotta simplify two sets of parentheses multiplied together, root three plus five and root three minus four. And to do that, we’re gonna multiply each term in the first bracket by each term in the second bracket like this. So that gives us root three times root three. And then we got root three times negative four. So we’re gonna be taking away root three times negative four, which we’re gonna write the four before the root three. So that becomes negative four root three. Then we’ve got positive five times root three, so that’s positive five root three.

And finally, five times negative four, which is negative 20. And looking at those, we’ve got root three times root three. Well that’s just three. Then we’ve got negative four of these root threes. And then we’re adding five root threes to that. So that’s gonna give us positive one root three or just positive root three, as we could write it. And finally, we just got a negative 20 on the end. So combining like terms, we’ve got three take away 20 is negative 17. And then, we’ve just got one root three on its own. So the answer is negative 17 plus root three. And like all of these questions, it doesn’t matter actually which order you write those terms in. So you could write root three minus 17. In fact, there’s a case for doing this because quite often we don’t like to lead with negative signs like that. So root three minus 17 is another perfectly good answer.

Our next question then is seven plus root five times seven minus root five.

Now, there are two different ways we can approach this. So we’ll do both of them, and then you can compare between the two. And the first method is to just multiply each term in the second bracket by each term in the first bracket. And that gives us seven times seven is 49. Seven times negative root five is negative seven root five. Then root five times seven, well I’m gonna do that the other way around. Seven times root five, so that’s positive seven root five. And lastly, root five times negative root five. Well, the square root of five times the square root of five is just five. And positive times negative makes negative, so that’s gonna be negative five. So we’ve got negative seven root five plus seven root five. Well, they’re gonna cancel each other out. So that’s gonna leave nothing in the middle. So we’re just left with 49 minus five, which is 44.

So multiplying out the parentheses like that, not massively difficult. And it’s given us our simple answer of 44. But let’s have a look again at the question. If you recognise this form, that is the difference of two squares. If you think back to your factoring of quadratics, 𝑎 squared minus 𝑏 squared, so one square number take away another square number, can be factored like this. 𝑎 plus 𝑏 times 𝑎 minus 𝑏. And if we let 𝑎 equal seven and 𝑏 equal root five, that would give us seven plus root five times seven minus root five, which is what we’re trying to do. So this is just 𝑎 squared minus 𝑏 squared. In other words, seven squared minus root five squared. Well, seven squared is 49, and root five times root five is just five. So we’ve got straight to 49 minus five, which is 44.

So as long you remember the rule of two squares, and you keep that kind of stuff in your brain. Then this kinda question can actually give you less working out to do this way then you had the other way.

Now, our next question, simplify six root seven minus four root two times six root seven plus four root two.

And if we look inside these parentheses, we’ve got six root seven in both places. And we’ve got four root two in both places. So we’re subtracting in one, adding in the other. This is just like the last question. This is a difference of two squares question. And this time, 𝑎 is six root seven, and 𝑏 is four root two. So we can rewrite this as six root seven squared minus four root two squared. And six times root seven all squared means six times root seven times six times root seven. And four root two all squared means four times root two times four times root two. And with multiplication, remember, it doesn’t matter what order we multiply these things together. So I’m just gonna slightly reorganise that.

So just swapping the order that I multiply some of those terms together means that I’ve got the radicals together and the rest of the numbers. So I’ve got six times six times root seven times root seven and four times four times root two times root two. Well, six times six is 36, and root seven times root seven is just seven. So I’ve got 36 times seven. And then, four times four is sixteen. And root two times root two is two. So this simplifies to 36 times seven minus 16 times two. Now, I’m sure you’re all familiar with your 36 times table. So 36 times seven is 252 obviously. And 16 times two is 32. Then, 252 minus 32, it gives us our answer, 220.

Now, our next question, simplify root five minus root six all squared.

Now, be very careful when you do these sorts of questions. The most common mistake is just people look at that especially in the heat of an exam and they think, “Oh, that’s root five all squared minus root six all squared.” And so they end up doing five minus six. Now that is completely wrong. You should always write it out in full; something squared is that thing times itself. So root five minus root six all squared is root five minus root six times root five minus root six. And we’ve got to multiply each term in the second bracket by each term in the first bracket. So that’s root five times root five, and then we’ve got root five times negative root six. That’s minus root five root six. And then we’ve got minus root six times root five.

Well, it doesn’t matter what order you multiply them together in. So I’m gonna put that as root five root six again. And it’s negative times positive, so it’s a negative again. And then, we’ve got negative root six times negative root six or negative times negative is positive. So looking at those expressions, we’ve got root five times root five, which is just five. And we’ve got root six times root six, which is just six. In fact, it’s positive six. Now, root five times root six is also the same as root five times six. So we’ve seen this before when we’ve been factoring out and simplifying these surds or these radicals. But it works the other way as well. So root five times root six is root five times six, and that is root 30. So we’ve got negative root 30 take away another root 30.

Now let’s look at the terms; collect like terms. Five plus six is 11 and negative root 30 take away another root 30 is negative two root 30. Now looking at the contents of this radical, 30. Let’s think about the factors of 30. Well, 30’s got lots of factors. But looking at them, only one is a square number. And that’s not gonna help us in this situation to reduce the size of the number. So we can’t factor it down any further to simplify that radical or that surd. So this is our answer, 11 minus two root 30.

Let’s move onto our second last question then.

Root seven time three root seven minus five minus two times four minus five root seven.

Well, we’re gonna use the distributive property of multiplication to tackle this question. So looking at the first parentheses first, we’ve got root seven times three root seven, which is the same as root seven time three time root seven. Then, we’ve got root seven times negative five. So we’re taking away root seven times five which is the same as five times root seven. Then, we’re gonna do negative two times four which is negative eight. So we’re taking away eight. And negative two times negative five root seven. Well, negative two times negative five is positive 10. So we’ve got positive 10 root seven.

Right, let’s look through here. Now, in this first term here, we’ve got root seven times root seven times three. Well, root seven times root seven is seven, and seven times three is 21. And we can’t simplify the second term, so minus five root seven. And we’ve got minus eight. And likewise, we can’t simplify the last term. Now, 21 take away eight is 13. And negative five root seven plus 10 root seven is positive five root seven. And since seven doesn’t have any square factors bigger than one, we can’t simplify this down any further. So 13 plus five root seven is our answer.

Lastly then, just for a bit of fun, let’s try this pretty tricky-looking question.

Simplify the square root of 15 plus the square root of 19 all squared times the square root of 15 minus the square root of 19 all squared.

Now, if you’ve been paying attention throughout the video, you’re probably spotting the fact, “Oh, that looks like difference of two squares!” But it’s not quite in that format, is it? So the difference of two squares says that 𝑎 squared minus 𝑏 squared is 𝑎 plus 𝑏 times 𝑎 minus 𝑏. And we’re nearly there. But we’ve got this problem of these two squareds above the parentheses here. So if we do a bit of rearranging then, we will be able to use that. So let’s have a look at it. Well, root 15 plus root 19 all squared is just root 15 plus root 19 times root 15 plus root 19. So that’s the first parentheses dealt with, and similarly for the second.

And now, we’ve got four sets of parentheses, all multiplied together. And when we multiply things together, it doesn’t really matter what order we do them in. So I’m gonna rearrange these. So I’ve just swapped around the order of the middle two sets of parentheses there. And that’s left me with root 15 plus root 19 times root 15 minus root 19 all times root 15 plus root 19 times root 15 minus root 19. Now each of those is the difference of two squares format that we were looking for. So if 𝑎 is root 15 and 𝑏 is root 19, we have this pattern here. And we also have it here. And that means that 𝑎 squared is root 15 all squared, which is 15. And 𝑏 squared is root 19 all squared, which is 19. And 𝑎 squared minus 𝑏 squared is 15 minus 19.

So let’s go back to our question then. We’ve got 𝑎 plus 𝑏 times 𝑎 minus 𝑏 times 𝑎 plus 𝑏 times 𝑎 minus 𝑏. And 𝑎 plus 𝑏 times 𝑎 minus 𝑏 is the same as 𝑎 squared minus 𝑏 squared. And as we just said, 𝑎 squared minus 𝑏 squared is 15 minus 19. And 15 minus 19 is negative four. So that becomes negative four times negative four which is positive 16.

So remembering our difference of two squares and doing a little bit of reorganising to get things in the right format for that has saved us an awful lot of multiplying out. And got us to a very simple answer of 16 instead of all of this stuff up here, which we started off with at the beginning of the question.

Okay, good luck with multiplying radical expressions then.

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