Law of indices

Hi, welcome to algebra and functions. In this session, we’re going to have a look at how to use the laws of indices to simplify expressions. So let’s go through the laws. There’s six in total. So, first of all, we have a of the power of n times a to the power of m equals a to the power of n plus m. So for example, if you have x cubed times x squared, we’ll have x to the power of 5 because 3 plus 2 is 5. Our next law is a division law. So if you’ve got a to the power of n divided by a to the power of m, you can subtract your powers.

So, for example, if you have y to the power of 7 divided by y to the power of 3, 7 take away 3 is 4. Next up, we’ve got the power law. So a to the power of n times– a to the n to the power of m is a to the power nm. So, for example, if we have x cubed squared, we multiply 3 and 2 together, which gives us x to the power of 6. Next up, we have negative powers. So a negative power means we do 1 over. So, for example, we have y to the power of minus 3. That means we are doing 1 over y cubed.

Finally, we have fractional powers. So anything where you have a power, the denominator of the fraction is what you root by. So, for example, if we had x to the power of 3 over 5, this means we are raising x to the power of 3, but we are also rooting it to the fifth root. Finally, anything to the power 0 is 1. So, for example, if we have y to the power of 0, that will give us 1. So let’s do some examples. So, first of all, let’s look at laws one and two. So question one, we’ve got x cubed times x to the power of 7. So we’re going to use our first law here.

So we add our powers together.

That’s x to the power of 10. Now, this one’s slightly harder because we’ve got a number in front of the x. We’ve got a coefficient of x. So what we need to do is we need to multiply 3 and 5 together. So three times 5 is 15 and then we can carry on and use our laws of indices as usual. So we’ve got x squared times x for the 4, so that’s x the power of 6. OK, let’s move on to our division law now. So when we’re dividing, we can subtract the powers. So 9 take away 3, that’s x the power of 9 minus 3 is x the power of 6.

Finally, again, it’s slightly harder because we have numbers in front. We have coefficients in front of our x’s. So we do 10 divided by 2, so that’s 5. Then we can deal with the x’s and apply our law of division. So x the power 4 divided by x to the power of 3 and we’re going to do 4 takeaway 3 to give us the 1. And, of course, we can write that as just 5x without the power of 1. OK, next set of examples. So we’re going to have a look now at the third law, the power law.

So when we’ve got x squared raised to the power of 5, looking at that law, that means we’re going to multiply those two numbers together. So x the power of 2 times 5 is going to be x to the power of 10. Now, this one is slightly harder because we’ve got a coefficient, again, in front of the x value. Now, if you notice, this is in brackets. So we are raising all of this 3x squared to the power of 3, so we have to do 3 to the power of 3 as well. So 3 cubed is 27. And then we can apply our law– our power law to the x squared.

So 2 times 3 is 6, so it’d be x to the power of 6.

Our third example, we’re now going to move on to our fourth law. So it’s our reciprocal law or our negative power law. So we want to put this x cubed up to the numerator, so we need to give it a negative power.

OK, next set of examples. So we’re going to have a look at laws five and six now. So first of all, law five. So this is a square root. So we can rewrite this as 5x to the power of 3 over 2. Because when we have a fraction as our power, the denominator is what we are rooting by. And because we are square rooting, the denominator is a two. What happens if we aren’t square rooting, though? So in this second example, we are doing the fourth root. So it will be x– this is raised to the power 5. So the numerator of the fraction will be 5, but the denominator this time will be 4. OK, third example.

Then we got x to the power of 0, so we’re having a look at our final law now. So anything to the power of 0 is 1, so that would just be 1. And then finally we’ve got 6 times x the power of 0. So x the power of 0 is 1. So 6 times 1 is just 6. So just be really careful. This power 0 is only on that x, it isn’t on the 6 as well. If we wanted it on the 6 as well, we would have to have brackets around the 6x.

OK, so just looking at a couple of harder examples now. So we want to rewrite these using our laws of indices when we’ve got quite a few laws going on. So, first of all, we want to move this x cubed square rooted up to the numerator. But then we also want to change this square root so it’s power of a half. So you can do it in any order. I’m going to rewrite it on the denominator first. So it’s 3– the power is 3 over 2. Now we can move that up to the numerator. So it’ll be 3x to the power of minus 3 over 2.

Finally, again, we’re going to move this x squared from the denominator up to the numerator. So we’re going to have 2x to the power of minus 2 all over 3. Thank you.