In this video, we're going to cover how to simplify expressions that contain surds, like this one here, where we have the square root of 125 minus 2 root 45 plus, in brackets, root 5 plus 2, all squared. First though, we're going to quickly recap the rules. The first is that when you multiply or divide surds, you just multiply or divide the numbers inside the root For example, root 5 times root 6 is the same thing as root 5 times 6, so root 30, while root 20 divided by root 10, which could also be written as a fraction, is the same thing as the square root of 20 divided by 10, which is just root 2. The important thing to remember though is that addition and subtraction don't work this way. For example, if you have root 13 plus root 6, you can't add them together to get root 19, just like you couldn't do root 13 minus root 6 to get root 7. This is because the numbers inside the surds are different, so you can't add or subtract them.
But if you had surds with the same numbers inside, like 2 root 3 plus 5 root 3, then you could add them together by adding together these numbers in front. so this would give us 7 root 3. And the same thing goes for subtraction, so 6 root 7 minus 2 root 7 would be 4 root 7. The last rule is that when you multiply a third by itself, like root 7 times root 7, the thirds disappear, and it becomes just 7. If it helps, you can think of it as the square root of 7 times 7, so the square root of 49 which is 7. So if we go back to our original expression, how would we simplify something like this? Well in a real exam, they'll normally add some words to the question, like write all of this in the form of a plus b root 5, where a and b are integers, and all this means is that we have to simplify the expression until it looks like this. so it might be something like 4 plus 3 root 5 or 7 minus 2 root 5. Basically a and b can be any positive or negative whole number. To do this we're first of all going to have to simplify each of these terms and then we'll need to add them all together.
Let's start with the square root of 125. If you're not sure how to simplify thirds it's probably best watch our previous video first which details exactly how to do it. To quickly recap though, the idea is to rewrite the cert that you're trying to simplify as a product of its factor pairs, and pick the one that contains the biggest square number. So in this case that would be root 5 times root 25, because 25 is a square number.
Then to simplify it, we actually take the square roots. So because the square root of 25 is 5, we're left with root 5 times 5, which we can rewrite as 5 times root 5, or just 5 root 5. So we've now simplified the square root of 125 all the way down to 5 root 5. Next, we have the 2 root 45, so 2 lots of square root 45. And to simplify the root 45 part, you can rewrite it as root 9 times root 5, because this is a factor pair, or 45, and 9 is a square number. Then because the square root of 9 is 3, you can simplify it again to 2 times 3 times root 5, so 6 times root 5, or just 6 root 5. And remember it's really a minus 6 root 5. because we were taking away the in the original equation.
The last term, plus 2 all squared, is a bit trickier, because it requires you to know how to expand brackets. The first thing you need to know is that when brackets have a squared symbol like this, it means that you have to multiply the entire bracket by itself. So you want to write two of these brackets next to each other, and then multiply everything in his first one. by everything in this second one. So root 5 times root 5, which is 5, plus root 5 times 2, which is 2 root 5, plus 2 times root 5, which is 2 root 5 again, and finally 2 times 2, which is 4. And if we simplify this, 5 plus 4 is 9, and 2 root 5 plus 2 root 5 is 4 root 5. So we end up with 9 plus 4 root 5. So if we now rub out all of our workings we can see that we've got the simplified form of all three of the terms in the question.
5 root 5 for the root 125 term, minus 6 root 5 for the minus 2 root 45 term, and lastly 9 plus 4 root 5 for the root 5 plus 2 all squared term. The final step is just to add these together, so 5 root 5 minus 6 root 5 plus 9 plus 4 root 5, which simplifies to 3 root 5 plus 9. However, because the question asked for our answer in the form of a plus b root 5, we want to quickly rewrite it as 9 plus 3 root 5, where 9 is the a, and 3 is the b. Before we finish, let's run through one more quick example, but without all of the workings this time. So for this question here, we're trying to simplify root 48 plus 2 root 75 plus root 3 squared, and put it in the form of a plus b root 3. So if we simplify root 48, we get 4 root 3, If we simplify 2 root 75, we get 10 root 3. And if we simplify root 3 squared, we just get 3. Then we can add these three terms together, so 4 root 3 plus 10 root 3 plus 3, and that will give us 14 root 3 plus 3. But remember, because we want it in the form of a plus b root 3, we're going to have to switch this around and rewrite it as 3 plus 14 root 3, which will be our final answer. Anyway, that's everything for this video.
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