In this video, we're going to cover how to rationalize the denominator of a fraction, like that of 4 over root 5, or 7 over 2 plus root 3. Now rationalizing the denominator is kind of a weird phrase, but basically all it means is that you have to make the denominator of the fraction a rational number. And the key thing to remember is that thirds aren't rational. Instead they're considered irrational numbers. numbers.
So in questions like this, they're really just asking you to get rid of any thirds that are in your denominator. For questions like 4 over root 5, where there's just one term in the denominator, all you have to do is multiply the top and bottom of the fraction by that third. So we'd multiply our fraction by root 5 over root 5. This works because 4 times root 5 is 4 root 5, and root 5 times root 5 is just 5, so we no longer have a third in our denominator.
For questions like 7 over 2 plus root 3 though, where the denominator has two terms, we still have to multiply top and bottom by the denominator again, but importantly we also have to change the sign. So these plus signs actually get changed to minus signs instead. And likewise if we'd started with a minus sign, we would have had to change it to a plus sign.
Next we need to multiply it all out. So for the top, 7 times 2 is 14, and 7 times minus root 3 is minus 7 root 3. Then for the bottom, 2 times 2 is 4, 2 times minus root 3... is minus 2 root 3, root 3 times 2 is plus 2 root 3, and root 3 times minus root 3 is minus 3. So if we simplify the denominator, we can cancel out the minus 2 root 3 and the plus 2 root 3, and just do 4 minus 3, which will give us 1 as our new denominator. Our numerator doesn't simplify down though, so we're left with 14 minus 7 root 3 all over 1, which is really just 14 minus 7 root 3. Let's try a couple more.
In this first one, we need to rationalise the denominator of 6 over root 3. So the first thing is to take the 6 over root 3 and multiply it by root 3 over root 3, which gives us 6 root 3, as the new numerator, and just 3 as the new denominator. And because it's a fraction, you should always see if it can be simplified at all, which this one can, by dividing top and bottom by 3, to get 2 root 3 over 1, or just 2 root 3. For this next one, we've got to simplify 7 plus root 5, over root 5 minus 1. and they're asking us to give our answer in this particular form of a plus b root 5, where a and b are positive integers. Now asking for the answer in the form of a plus b root 5 is really just their way of telling you that you're going to have to rationalize the denominator, because we can see that in a plus b root 5 there isn't any third on the bottom, and all they mean by positive integers is that a and b need to be positive whole numbers.
So the first step is to take our whole fraction and multiply it by root 5 plus 1 over root 5 plus 1, because this is the same as our original denominator, but we've changed the minus sign to a plus sign. Next, you need to actually do the multiplication. So for the numerator, you should get 7 root 5 plus 7. plus 5 plus root 5, which simplifies to 12 plus 8 root 5, and for the denominator you should get 5 plus root 5 minus root 5 minus 1, which simplifies to just 4 because the positive and negative root 5 cancel out and we're left with 5 minus 1 which is 4. And then you can simplify the whole thing.
by actually dividing the numerator by this 4. to give us 3 plus 2 root 5 as our final answer, and that's already in the form of a plus b root 5 like they asked us for. Anyway, that's everything for this video, so hope that made sense, and cheers for watching!