A surd is an expression that includes a square root, cube root or another root symbol. Surds are used to write irrational numbers precisely – because the decimals of irrational numbers do not terminate or recur, they cannot be written exactly in decimal form.
We are looking at thirds. So we'll first focus on the three main rules that we use to basically tackle all thirds questions. We'll then look at the subtopics that are involved in thirds and then finish off with some exact questions at the end. So before we start, let's just make it super clear. Says is really just another name for roots, right?
The square of X is set or the cubed roots of eight is a third. Or the 17th root of A plus B is a third. Anytime we see this root sign, we're dealing with thirds. So let's start off with the three main rules. So the first rule is really just the definition of the square root, right?
The square root of anything. Square of anything. Let's pick nine. As an example, we know the square of nine times the square of nine. By definition, this must give us nine, right?
The square root tells us that the square root times itself gets us the original number. It gets us that number inside the square root. It gives us nine. In this case, the square of nine is three, right? We can double cheque this three times three does in fact give us nine.
And we can generalise this. Rather than using just nine as an example, we can generalise this. And we can say the square of any number times the square root of. Again, that same number inside the square root has to get us back to that original number has to give us A. And that is our 1st 3rd rule.
The second rule is even a more general version of this. Really. It says that if we've got square root of some number times the square root of a different number, we can smush these two things together and put them both inside the square roots. And now we could double cheque this if we say A and B are the same. This also says that root A times root A is the square root of A times A.
Well, we know square root of A times A times A is just a squared, right? The square root of A squared. Well, if we square something and then immediately square root it, we're just undoing what we just did, right? The square and the square complete opposites. It's like timing by five and dividing by five.
They're complete opposites. What we end up with is our original number. We get back to a we can kind of almost relate these two rules together just by replacing B with A. But this is kind of even more general version of the one above. So this is our first rule.
This is our second rule. Our third rule is again, not too dissimilar to the second rule. If we say rather than route a times root B, I say this is root A times. Let's say it's times one over.
Well, if I use this power rule, I'm actually going to say this is the square root of one over B.
If I use this above power rule, I know this is going to give me the square root of A times one over B is the exact same as A divided by B, right? 20 times a quarter is five or 20 divided by four is also five. So I can either times by one over B or I can divide by B. And likewise with this I can either times by the square root of one over B or I can divide by just the square of B. And from that very simple substitution we get our third rule.
The third rule says root A divided by root B is root A divided by B. Let's look at adding and subtracting. So this is relatively simple. If we've got as an example, let's say we've got two lots of something plus five lots of something. Well, we know that's going to give us seven lots of that same thing, right?
This could be apples or oranges or vases or anything we want to see. Or we could use algebra. We could put x in here. We could say two x plus five x equals seven x. Or we could even put thirds in here.
We could say two root three plus five root three gives us seven root three. This is basically the way we tackle adding subtracting, right? If we were to generalise this a lot of some square root or something, plus some other lot of the square root of that same thing. What we do is in this case we added two and five together, right? We add our two numbers together, we do A plus B and we multiply our original set, our original route by that resulting product.
That's basically the rule that we use to tackle adding and subtracting, right? It doesn't have to be adding. We can put plus or minus in here. This rule still stands. I'm going to take this down here and we'll look at multiplying and dividing.
So there's one trick that we can use to simplify things massively for ourselves. If we look at an example, let's say we've got two root twelve times five root three. Let's see what we get with this. Well, two, route twelve. This is just a shorthand way of saying two times route twelve, right?
And we're timing this by five times route three. What we've got here is we've got four terms all timing each other together in a long line. And when we've got this, it doesn't matter which way around, we do this multiplication. I could flip this all and I could go from the right to the left or from the left to the right. Or I could do five times twelve times three times two.
It doesn't matter what order we do things in. So we can use this to our advantage, right? We can simplify things massively and we can instead group our numbers together. So two times five at the start and we can group our search together at the end. In this case it's root twelve times root three.
Well we know two times five is ten. We know root twelve times root three. We can use our 2nd 3rd rule to our advantage and we can say this is the exact same as the square root of twelve times three, which is the exact same as the square root, the exact same as square root of 36, which is the exact same as just six, isn't it? What we end up with is just ten times six and that gives us 16. So that's what we can use, the very simple principle of rearranging things in a long line of multiplication to our advantage, simplify the problem down and we end up with a nice number at the end.
And the same goes for dividing, right? We can always reorder things in a division as well, brackets for example, let's say we've got root five, brackets five. What we do is what we normally do, we multiply out, we do root five times t and root five times seven, root five or root five times t is just and our second term is going to be root five times seven, root five. Well, I write this down to start with. Root 55.
We can rewrite this though we can again use this fact that we can reorder this multiplication and we can write this as seven.
Again, we know from our first rule, from the fundamental definition of what a square root is, that root of anything times of that same thing gives us the original number without the square root. So root five times root five I know is going to give us five. So this just becomes seven times five, which is just 35 and we end up with two root five plus 35 for their answer. Finally, rationalising the denominator. Let's say we've got eight over root two.
By rationalising, what we mean is we want to get rid of this root sign. We don't want to have any roots in our denominator if we can avoid it. So what we're going to do is we're going to use a trick again, we're going to use the exact same first rule, we're going to use the fact that we can take a root times it by itself and it will get rid of that root. We're going to do that same thing here. We're going to take this root t in this case and we're going to times it by root t and obviously we have to times the top by root two as well to keep that whole value the same.
What that gives us is eight root two on the top over t times root t on the bottom. And we know from our first rule and the definition of what square root is that this is the exact same as eight root two over just t and we can simplify further. Eight over T is going to give us four.
Right, let's look at some exam style questions to finish off with. So let's do questions one, right? Route 48 in the form of Kroot three. And so we know that we're going to have to use our third rule, right? We're going to have to write route 48 as something times something A times B in this square root.
We're just going to figure out what A times B has to be. Well, I know that I've got a root three in my final answer, so I know that B has to be three. Well, something times three has to give me 48. And if my multiplication is fine, that's going to give us 16. So that tells me that it's the exact same as 16 times three inside that square root.
I can use my second rule which says I can split multiplication. And this is the exact same as route 16 times root three. We know route 16, it's just four to end up with four times root three. And that's the answer. We get four roots three.
Let's see question five, expand and simplify. So we'll do the same thing we always do. We should get four terms. We'll do two times two. First of all, we get four.
We do two times minus root three. We get minus two, root three. We then do root three times two and we get plus two roots three. And we do root three times minus root three. We get minus three.
Well, that and that are going to cancel out. They're just opposite to each other. We end up with just four minus three, which gives us one. So one is going to be our answer. Rationalise the answer.
This is very similar to what we just did on the previous page. So we're going to use this trick again that the square root of three times square three is going to cause the square root to vanish, right? We do the same thing to the top because we have to do the same thing to both the top and the bottom. Always this then the cons six roots three on the top and root three times root three gives us just three on the bottom six. Root three over three is going to give us just two routes three as our final answer.
And that is the end of set.
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