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Video: Multiplying Radical Expressions

Tim Burnham

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.

17:15

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 forty-five 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 forty-five.

And now we’re going to try and find the biggest square factor we can find of forty-five. 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 the results that we get are square numbers. And the first one we encounter is also gonna be the biggest square factor. So forty-five divided by two doesn’t work; forty-five divided by three is fifteen, but that’s not a square number; forty-five divided by four, doesn’t go into forty-five exactly; forty-five divided by five is nine. And that is a square number, so that is our largest square factor so I’m gonna rewrite forty-five as nine times five.

And now root nine times five. We’re gonna split it 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 fifteen, and the square root of five times the square root of five is just five, and fifteen times five is seventy-five, so the answer is seventy-five.

The next question then is express four root seven times two root forty-nine 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 forty-nine means two times root forty-nine. But also forty-nine is a perfect square, so root forty-nine 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 fifty-six, so that’s fifty-six 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 fifty-six 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 fifteen; and five times two root five, that means five times two times root five and five times two is ten so that’s ten root five, so we got fifteen minus ten 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 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 fifteen. And the square root of three times the square root of three is just three. So this gives us six root three plus fifteen times three. Well fifteen times three is forty-five. So it’s important to take note of the order of operations, but this all simplifies down to six root three plus forty-five.

Now our next question is littered with negative signs so we’re gonna 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 going to 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 twelve. 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 twelve So we’ve ended up with twelve root two take away negative twelve. Well if we take away negative twelve, that’s the same as adding twelve. And root two won’t simplify anymore, so this is our answer: twelve root two plus twelve.

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’s gives us root three times root three, and then we’ve got root three times negative four. So we’re gonna be taking away root three times negative four. Well actually, 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 twenty. 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 twenty on the end.

So combining like terms, we’ve got three take away twenty is negative seventeen, and then we just got one root three on its own. So the answer is negative seventeen 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 seventeen. 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 seventeen 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 forty-nine. Seven times negative root five is negative seven root five. Then root five times seven, well I’m going to 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 forty-nine minus five, which is forty-four.

So multiplying out the parentheses like that, not massively difficult, and it’s given us our simple answer of forty-four. 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 takeaway 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 will 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 forty-nine, and root five times root five is just five. So we’ve got straight to forty-nine minus five, which is forty-four.

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 this 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, that 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 thirty-six and root seven times root seven is just seven, so I’ve got thirty-six times seven. And four times four is sixteen, and root two times root two is two.

So this simplifies to thirty-six times seven minus sixteen times two. Now I’m sure you’re all familiar with your thirty-six times table so thirty-six times seven is two hundred and fifty-two obviously and sixteen times two is thirty-two. Then two hundred and fifty-two minus thirty-two, it gives us our answer: two hundred and twenty.

Now our next question: simplify root five minus root six all squared. Now be very careful when you do these sort 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 thirty.

So we’ve got negative root thirty take away another root thirty. Now let’s look at the terms; collect like terms. five plus six is eleven and negative root thirty take away another root thirty is negative two root thirty.

Now looking at the contents of this radical, thirty. Let’s think about the factors of thirty. Well thirty’s got lots of factors, but looking at them only one is a square number and that’s not going to 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: eleven minus two root thirty.

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 to 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 time four which is negative eight. So we’re taking away eight. And negative two times negative five root seven or negative two times negative five is positive ten, so we’ve got positive ten 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 twenty-one. 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 twenty-one take away eight is thirteen and negative five root seven plus ten 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 thirteen 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 fifteen plus the square root of nineteen all squared time the square root of fifteen minus the square root of nineteen 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 is 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 fifteen plus root nineteen all squared is just root fifteen plus root nineteen times root fifteen plus root nineteen. 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 going to 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 fifteen plus root nineteen times root fifteen minus root nineteen all times root fifteen plus root nineteen times root fifteen minus root nineteen. Now each of those is the difference of two squares format that we were looking for. So if 𝑎 is root fifteen and 𝑏 is root nineteen, we have this pattern here and we also have it here.

And that means that 𝑎 squared is root fifteen all squared, which is fifteen, and 𝑏 squared is root nineteen all squared, which is nineteen, and 𝑎 squared minus 𝑏 squared is fifteen minus nineteen. 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.

Now, and as we just said, 𝑎 squared minus 𝑏 squared is fifteen minus nineteen. and fifteen minus nineteen is negative four. So that becomes negative four times negative four which is positive sixteen. 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 sixteen 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.