Express root 128 plus root 18 minus 8 root a half in its simplest form.
In order to express this in its simplest form and solve the problem, we’re actually gonna have to simplify each term, and I’m actually gonna do it one by one so we can see exactly how we do that. Starting with root 128, simplify root 128.
What we need to do is we actually need to find two factors that multiply together to give us a 128. And the key tip here is, when finding the two factors of a surd, check that one is the highest square factor that goes into that number. So therefore, root 128, we’ve got root 64 multiplied by root two.
And the other reason that works is the little rule we’ve got here on the right-hand side is that root 𝑎 multiplied by root 𝑏 is the same as and equals root 𝑎𝑏. As I said, we need to check for the highest square number factor, which in this case would be root 64. So if you didn’t actually use root 64, you could’ve got this, which was root 16 multiplied by root eight; this isn’t incorrect and you can still work through it and get exactly the right answer. However, there are more steps, and with more steps, there’s more chance for an error. So follow the tip and it’ll actually make your life definitely easy with these kinds of questions.
Okay, now to solve this one, we’ve got root 64, which gives us eight, so we get eight multiplied by root two, but with convention, the way we write that is eight root two. Now we can move on to root 18. Again, following the same method, we get root nine multiplied by root two, which gives us three root two.
And again, another little tip here, and this is an interesting one: whenever you’re doing a problem like this and it’s asking you to simplify, look out for the common surd. So here they’re both root two, and that’s what you should be ending up with, because to enable you to simplify, we’re gonna have to have a common surd involved. So that’s a good little tip. If you haven’t got that, so if you came up with something like root six or root eight, then have another loo- look at your working out and see where you might’ve gone wrong.
Okay then we’ve got a final term, which is eight root a half, and this we’re gonna do slightly differently. Okay well, first of all, we can say that eight root a half is the same as eight root one divided by root two, which we’re obviously using again the rule on the bottom right-hand side this time, which now gives us eight and one over root two, which can be rewritten as eight over root two.
Well we now have got root two as our denominator, and we don’t want that; we don’t want a surd as the denominator, so what we need to do now is rationalize the denominator. And in order to rationalize the denominator, we actually multiply the numerator and the denominator by root two. And the reason we do this is because root two multiplied by root two will give us two and therefore remove the surd from the denominator of our term.
So this gives us eight root two over two, and then we can simplify further by dividing the eight by two, which gives us that fully simplified term of four root two. And again, great, we can see that it’s root two, so we’ve got that common themed root two following through, which is now gonna help us when we’re gonna simplify.
So we’ve got our root 128 plus root 18 minus eight root a half, which is gonna give us eight root two plus three root two minus four root two. Great! We can now just simplify these like algebraic terms, which gives us our final simplified answer of seven root two, and that’s because we’ve got eight plus three, which gives us 11 root two minus four root two gives us seven root two.