In this video, we look at the Laws of indices and give rules for simplifying calculations or expressions involving powers of the same base. We have base numbers and indexes or powers when writing certain formulas. For example; 2^{4}. 2 is the base number and 4 is known as the power or index number.

We are looking at laws of indices. So we'll first look at what we mean by indices. We'll focus on the five main power rules and then we'll finish off with a couple of exam style questions. So indices, just the plural rule of index and index is just another word for power. So this could be laws and indices or laws of powers.

They are interchangeable. So what do we mean by powers? Well, just to make sure we're all on the same page, if I have, say, three to the power of four, all that means is we've got four threes all in a long line tasing each other together. So that would be three to the four. Likewise, if we had say, I don't know, seven to the five, all this means is we've got five seven all in a long line, XD each other together.

So that's what our powers represent. Let's look at our first power all. Now move this over slightly to give myself some more space. There we go. So X to the A times X to the B.

Well, we talk about x to keep things general. So it's some number to some power times that same number to some other power. Let's do an example. Let's say we've got eight to the power of three times eight to the power of four. Well, all we're doing here is let's kind of draw this out and see where we're at.

So we know eight to three or eight cubes is just three eight all in a long line timing each other together. And eight to the four is just four eight in a long line, timing each other together eight times eight times a times A. And we're timing these two things together. Well, all we're left with is a much longer line of eight times each other, right? So in this case, we've got seven eight times each other together.

So we can rewrite this kind of working backwards. We can rewrite this as eight to the power of seven, right? This tells us we've got seven eight all in a long line time to each other together. And so, in general, our power rule can be x to the A times x to the B is equal to or what have we done here? We've just added up that all the eights that we've got.

We've got three eight here and we've got four eight here. We've added them up to get seven eight and this is eight to the seven. So in general, all we're doing is we're just adding up all of the in this case, to keep it general. We've got all the x's, right? We're just adding up these two numbers together to get our new power.

So that's our first, our first power rule, our second parallel, x to the A divided by x to the B. So let's keep these same numbers, right? Let's say we've got eight to the four. I'll do eight to the five, make it more interesting. Eight to the five divided by eight to the three.

Well, I'm just going to rewrite this as a fraction, right? We'll say this is eight to the five over eight to the three or eight cubed. And if I was to write this out, if I was to write this out to actually see what's going on here, well, eight to five tells me I've got five eight in a long line x each other together, and eight to the three tells me I've got three eight in a long line, each other together. What? We can see what a common factor of eight, right?

Whenever I've got a common factor, I can cancel it out from the top of the bottom. And likewise, again, I've got a common factor of eight on top of the bottom. And again, I've got a common factor of eight on top of the bottom. And so, because we basically cancel as many eight as we're dividing by, in this case, we're dividing by eight times itself, three times. And so we cancel out three eight.

We're basically knocking these off by a numerator. And so rather than eight to the five, or rather than five eight times each other together, we're left with just two eight times each other together. And so our new power, in this case, would be eight star of two, right? So in general, if we've got some number to some power and we divide that to the same number to some other power, because we end up cancelling the number of eight in this case that we're dividing by, we basically just end up subtracting them from our numerator. We subtract them from this eight to five.

So we end up just doing a very standard subtraction of powers.

That's our second power. Let's look at our third power. So this is X to the A, all to the power of B. So in this case, we know anything to some power. Let's say, for example, let's say we've got a squared to the power of two to the power of three.

Let's say. Well, I know a squared in this case is just eight times eight, and we're doing this power three. Well, the power three tells us we have three of these brackets in a long line timing each other together. And so in this case, I've got three of these brackets in a long line, and they're all timezone each other together.

There we go. Well, if you look at this, I don't really need these brackets anymore. I could rewrite this as just eight times eight times eight times eight times eight. Well, let's take a step back and see where we're at, right? So in this case, this two tells us that we've got eight times eight in our brackets.

It tells us we've got two eight per bracket, right? Two eight here, and we've got two eight here as well. The three tells us how many brackets we've got. The three tells us we need to take our bracket and times it by itself three times. So it's three tells us we've got three brackets.

And so if we wanted to work at how many eights we've got in total without actually counting up all of our eights, we just take the number of eights per bracket and we times it by the number of brackets. In this case, all we're doing is we're taking the number of eight per bracket, which is two, so it's going to be eight to the two, and we're timing this by the number of brackets. So the final power is going to be two times three, which in this case is eight to the six. Now, we can cheque that here as well. We've got how many eight, we've got 123-4568, all in a long line timing each other together, which again is eight to the six.

And so, in general, our power rule is going to be anything to some power to some other power. What we do here is we times the powers together, we times those powers together, and that allows us to calculate how many X's there are in a long line timing each other together. That is our third power rule, our fourth parallel. Try and fit it all in our fourth parallel, x to the minus A. Well, it takes a bit of a stretch of the imagination to start off with understanding this, but we'll see how we go.

So stay with me for 30 seconds. Let's say I've got eight to the five, divided by eight to the three. Well, in this case, we know from our previous power rule that we just subtract the two powers, right? So I end up with eight to the five minus three, which is eight to the two. I could rewrite this as well in a slightly different way.

Rather than dividing by eight cubes by eight to the three, I can time by one over eight to the three. And this isn't any different from what you're used to before. We could think of this as nicely, for example, we can do, say, 20 divided by four and we get five. Or we can think of this as 20 times one over four, and again we get five. So that's all I'm doing here.

I'm just rewriting this in a slightly different way. And we can also see what we get if we do, for example, eight to the five, times eight to the minus three. What we know from our first power rule, that when we times these two things together, we add the powers. And so we end up with five plus minus three in our new power, which gives us eight to the two. Well, if we look at where we're at now, we've started off with eight to the five in both cases, and we've ended up with eight to the power of two in both cases.

Or eight squared. So what we can see is that the middle things that we're timing by these two things must be the exact same. They must be the same thing. There's no way these could be different things. And so I can say, I know that eight to the minus three is the exact same as one over eight to the three is telling us to do is one over.

Basically, in general, we can say X, the minus A is just equal to one over XDA. That's our fourth power rule. Our final power rule is X to one over A. If we explore what that means, we can do again, stay with me for a second. Let's say we've got the square root of X all squared.

Well, I know if I square the square root, I get back to the original number. The square root of nine is three. If I square that, I get back to nine. And so that just gives me X. Likewise, again, stay with me for a second.

If I do say X to the half, the power of T, what I know from my third power rule that I times those two numbers together, right? So I times those two numbers together, I get half times T, which again is next to the one, which is just X as what we can see, right? We've if ended up with the same number, the same thing in both cases. We've got X in both cases and we started with root X here and X at the heart here, and we squared them both. Well, these two things must be the same thing.

Again, I know that X to the half must be the exact same as the square root of X. And am I right? This property is the two there is it the square root of X. And in general, I know that anything to the one over A is just going to be equal to the 8th root of that thing. That's our fifth and final power.

Let's look at a couple of examples now. So let's look at a past big question. So it's going to test our multiplying and dividing. We've got three to four times three to five and divided by three to two. Well, time zone.

I add the powers together. So I get three to the nine on top and dividing, I subtract the powers, so I end up with three to seven, three to the minus three. Well I know that minus just means one over, right? So three to the minus three just means one over three to the three. Three cubes is 27, so I get one over 27.

Finally, 64 to the two over three. I'm going to be slightly smart about this. I'm going to make this slightly easier and split into two bite sized chunks. This is like 64 to a third times two, right? Times two is just two over three, two thirds.

I know that from my previous power rule when I've got something that looks like this, I can work backwards, right? If I've got a times B in my power, I can write this as X to the A or to the power of B. So in this case, I can rewrite this as 64 to the third.

All to the power of 2. Well, 64 to the third. I know my cube numbers, so I know that's going to be four times four times four is 64. Power of two, give me 16.

That is the end of our laws of indices.

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