[music] [applause] right, so we can either look at fractional and negative indices when they’re put together so grab a piece of paper grab a pen make some notes and we’ll see what we make of these ones? So this first one we’ve got 25 to the power of negative 1/2!
Now, when it comes to these powers, we just need to remember what each piece does? So if we make some notes on this, the negative part of the power, as we’ve seen before, does the reciprocal it flips it over so flips it over the number on the bottom underneath underneath that line now, which i like to refer to as ground level. The number underneath does the root, so in the case for two that would be a square root and if it was a 3, it’d be a cube root and so on, and the number on the top is just a normal power.
So these are three things that we have to look at when we have these combinations going on, and sometimes we might not have a number on the bottom. Sometimes we might not have a negative, but if we have them all with to think about all three of these pieces, so doesn’t matter what order you do it in now, i tend to always go for this number on the bottom. First, purely because it’s easier to do the root of a number than it is to then do that needs to do the normal power when the numbers quite large. If we imagine there was a normal power of two on the top it’d, be a lot easier for us to square root this than it would be to do say, 25 squared first, so i always do the number on the bottom?
First, that’s just personal preference so that you can do it in any order that you like.
So what i’ve got to do here? first of all, i’m going to deal with this number on the bottom, so i’m going to go for this root first! So if i do the square root of 25 we’ll get the answer 5. So that’s the root dealt with now i’m going to move on to the other piece! So i’ve got the flip going on or the reciprocal! So if i flip 5 over remembering 5 is 5 over 1, if we flip that over it becomes 1 over 5 now the normal power there is just a 1 and a normal power of 1. Let’s find a different color fuse quite a lot now, a normal power of 1 doesn’t actually change anything and ^ 1 is just itself so that one over five there would be my final answer as have a look at another one, though, where that normal power on that top is something different to one okay. So here we go write down the value of 64 to the power of negative 2/3, so 64 to the power of negative 2/3! So, as you can see, we’ve got sir – on the top now so that – on the top as a normal power, it’s gonna square, whatever number we’re looking at okay as a normal power, the 3 on the bottom is going to do a root and specifically that’s going to be a cube root, because it’s a 3 and the negative symbol again is just going to flip. This number over do the reciprocal so we’re doing whatever all do we like i’m gonna, do the cube root first, as i said before, i always do the root to start with so i’m going to deal with this bit to start with? So if we do the cube root of 64, the cube root of 64 is 4 4 times 4 times, 4 is 64, so that is my cube root dealt with now. Let’s have a look at the next part, rather than flipping it over. Let’s square it. Let’s do this one to start with so 4 squared 4 times. 4 is 16, so 16 and the final part here, let’s do the flip, the reciprocal, so 16 is 16 over 1. Let’s flip that over 1 over 16 and there’s our final answer. So we have three steps there. I first did the cube roots, then i squared it and i flipped it a week again, you can rearrange the order that you do it in, but personally i’d, rather not do 64, squared and try and work out the cube root of it! I’d rather work out the cube root of it. While it’s a smaller number, so i would look at one more okay, so write down the value of 9 to the power of 3 over 2 or three halves so 9 to the power of 3 over 2. Now you might notice straight away: we’ve not got a negative symbol. Well, that’s fine! that just means we’re not going to do any flipping over we’re not going to write the reciprocal but i’m still going to deal with these just like i did before i’m, just not going to have to flip it over so on the bottom. We’ve got the root and those are two. So it’s a square root and on the top, we’ve got the normal power and that’s going to cube it to the power of 3, so 9 the square root of 9 to start with, if i deal with this root to start with the square root of 9 is 3 and then moving on to the normal power there.
The cube i’ll do three cubed, which i can write separately down here, three cubed 3 times 3 is 9 times, 3 again 27! So my final answer: there is 27, remembering i don’t have to flip it over because there’s no negative in this particular power is some fuse ever go up. Ok, so there’s four questions: just pay attention to what pieces are in the power, remembering the negative will flip it over the number underneath or do the root and the number on the top is the normal power. So have a go pause. The video we’ll go over the answers in a sec, okay, so 36 to the power of negative 1/2? So there is a negative, so it’s gonna flip and there’s a 2 on the bottom. So the first thing i wanna do is i’m going to do the square root of 36, which is 6 and then i’ll flip it over so 1 over 6 is our final answer there! What’s the one below 27 to the negative 2/3, so it’s going to flip it’s going to do a cube root on the bottom and the two?
So it’s going to square as well so i’m going to do the cube root. First cube root of 27 is 3? Then i’m going to square that. So we’ll square that which is 9 and then we can do the negative part! We can flip it over so 1 over 9 and there’s that one, the next one write down the value 100 to the power of 3 over 2. So no negative, but we are.
We have got a 2 on the bottom, so we’ll do the square root first and the square root of 100 is 10 and then we’ve got a 3 on the top! So we need to cube that 10 cubed is 10 times 10 times 10 10 times. 10 is a hundred and then times it by 10 again is a thousand, so 10 cubed is a thousand. So my final answer: there is 1000 and now on to the last one write down the value of 16 to the power of 3/4. Now we’ve got a 4 on the bottom now which we can look out and one of the other ones so a 4 on the bottom? Well, we have fourth root, so i need to do the fourth root of 16, so i need to know what number times itself 4 times makes 16. Now, when it comes to 4 through these numbers are never getting it very large because by the time we get to the far youth to the power of 4 or at 625 we get quite large, so we’re only looking really at 1, 2, 3, 4 and so on. So let’s have a thing: 16 is 2 times two times two times two, so the fourth root of 16 is 2. There you go, you can always just test that out down the bottom, just to make sure you happy with it.
So the 4th root of 16 is 2, it’s a 3 on the top, so we need to do 2, cubed and 2 times 2 times.
2 is 8 and that’s our final answer there. Okay! so there the answers. Let’s our backers, are slightly different ones. Okay, so write down the value of 9 over 25 is a fraction all to the power of negative 1/2. So we’ve got a fraction here now to the power of negative 1/2! We’re gonna treat it in exactly the same way, so we’ve got 9 over 25 and that’s all to the power of negative 1/2! So again, just applying those same rules. We’ve got a negative in the power, so we’re going to flip it! We’ve got a 2 down the bottom, which is going to do the root i’m gonna normal power on the top. There’s i’m not going to do anything for that, one on the top, so i’m just gonna label those two. So let’s decide what we’re going to do! First, let’s do the root first, as that’s what i keep doing? i keep doing this root first, so we’re gonna do the square root of 9 on the top and the square root of 25 on the bottom and the square root of 9 is 3 and the square root of 25 is 5, so we get 3/5, but then we’ve got one more step. We’ve got this flip still, so we still need to flip it over. So when we flip it over, we get 5 over 3 and that’s fine there? As a final answer. Just remembering, though, we could write our answer in a different way here, because that is an improper fraction! There we’ve got top-heavy, so we can think how many threes go into five and that’s one with a remainder of 2! So a remainder of 2/3 left over so i could also write my answer as 1 and 2/3 of thoughts write it as a mixed them, but but both answers.
There are fine, i thought. Look at another one! okay, here we go so write down the value of 8 over 27 to the power of 2/3, so we’ve got 8 over 27 and our power there is 2/3 so again, no negative in this one! So we’re not gonna have to flip it, but we do need to deal with these two numbers so on the bottom, i have a 3 which is a root, and that is a 3 there. So that is a cube root and on the top we have our normal power of 2, which is going to be to square the numbers all right so again, just like before you can do this and whatever or do you like, i’m gonna, stick with the route to start with, so i need to do the cube root of both these numbers and some ever hint in the numbers and they’re, both cubed numbers? So the betty, you know you’ve squaring the cube numbers easier. This becomes so the cube root of 8 is 2 and the cube root of 27 is 3, so we’re at 2/3! At this point now the next thing i’m going to have to deal with this this square, so we need to square both of them, so 2, squared and 3, squared and 2 squared is 4? Then 3 squared is 9. So my final answer: there is 4 over 9 i can’t leave them out so just like that it doesn’t simplify? So that is absolutely fine?
As a final answer. Ok, so some piece ever go up. Ok, here’s 4 questions have a go? Just remember negative flips. It number on the bottom does the root and the top number is a normal power. Okay, so pause the video there and we’ll go over the answers. Ok, so the first one we’re gonna have to flip that over and up two on the bottom does the root. So if i do the root to start with and i always do write down the working out here to show you what i’m doing the square root of 64 is 8, the square root of 81 is 9, and then we have to do the negative power so flipping it over 9 over 8 and again we can leave arendt’s like that or we could convert into a mixed number. 8 goes into 9 once with a remainder of 1, so 1 and 1/8 there. You go one on 1/8, that’s the next one below write down the value of 8 over hundred 25 to the power of negative 2/3, so 3 on the bottom. So we get a cube root both of these the cube root of 8 and the cube root of 125. So the cube root of 8 is 2 and the cube root of 125 is 5? Now we need to deal with that power on the top!
So what have we got? we’ve got a power of 2, so these are both gonna get squared, so 2 squared is 4 and 5 squared is 25?
We’re almost finished! We just need to finish this off now by flipping it over, because we’ve got that negative power again so flip that over and we get 25 over 4, which again we could write as a mixed number. 4 goes into 25 up to 24, so it goes in 6 times for the remainder of 1, so 1/4 dave-o could say six and a quarter as well onto this top right, so 25 over 100 to the power of three over two. So no negative! So we’re not going to flip it there’s a two on the bottom. So first things. First we’ll do the square root of both these numbers square root of 25 is 5 and the square root of 100 is 10. 5 over 10 is actually a half, so we can actually simplify this now. But let’s have a look: let’s just follow the process so 3 on the top, so we’re going to cube both of these so keep the 5 cube of 10, a cube of 5 is 125 and the cube of 10 is a thousand so again, i think this actually might have been easier for us to actually simplify that 5 over 10 to start with and then to keep the numbers from there because from here we really should simplify this.
This is our final answer, but we can actually simplify this so the top and bottom both divided by the both of our 125? Actually, that’s, not very nice! Let’s divide by 25 to start with so divide. The top by 25 gives us 5 and divide the bottom by 25? 25 goes into 100 four times, so that’s 5 over 40 and again that simplifies again so divide the top and bottom by 5. We get 1 over 8, okay. So our final answer there is 1 over 8!
Well then, we could leave our answer as and in 25 over a thousand is always best for still look to simplify there? I actually think, and if we pull this to the side, if we’d have simplified here, we would have got 1 over 2 and then we would have cubed the top keep the bottom and we just straight away.
Just got 1 over 8 and i had to worry about simplifying later on and to the last one 64 over a thousand to the power of 2/3 3 on the bottom. So we’re going to do the cube root of both these numbers, so the cube root of 64 on the top, the keep root of a thousand on the bottom, and we can work those 1. So the cube root of 64 is 4 aqib root of sorry.
There should be a thousand on the bottom. The cube root of a thousand is 10 and again, actually we could simplify this before squaring it. So i look at it in both ways.
So if i simplify it first, they both divide by 2?
That’s two fifths so from here. If we do the one on the top, we need to square them both. So we get 16 over a hundred and if we square it after we’d simplified it we’d have got four: that’s not a four for over 25 and that bad doesn’t simplify but 16 over 100. Does they both divide by 4? so if we divide the top by 4, we get 4 and the bottom by 4 we get 25. So our final answer: there is 4 over 25, but we could have got 16 over 100. It doesn’t actually ask us to simplify it, but i always look to simplify the way possible right, so that stays finished! That’s the end of this one. So if you like that video, if it was helpful, don’t forget to like comment and subscribe and i’ll see you on the next one! !