 I'm Lucy, and in this video we're going to look at the laws of indices. You should already know what indices are, but if you've forgotten, watch this video first. The laws of indices make solving complex problems involving powers much easier, and they are also essential for understanding a lot of algebraic processes. So there are six laws that we need to know, multiplying and dividing with indices, raising a power to a power, what a power of zero means, and then fractional and negative indices. We're going to look at the first four in this video, and then the last two in part two. I promise you they really are quite logical. Right, so multiplying indices. Look at how these are simplified. What do you notice? Simply you can see that when we multiply indices, we just add the powers together. So three plus four is seven, because really it is two three times multiplied by two four times, giving us seven twos, and three plus two plus one is six. Looking at the final example, see how we treat the numbers and the indices differently. So we do four multiplied by five first, and add the indices second. A really important thing to note is that the base has to be the same. So here, because they are both base two, we can add the three and the four, but on this one, we can't do anything, because one is base two and one is base three. Where do you want to go next? So what happens when we divide with indices? Have a look and see if you can work it out. When we divide indices, we subtract the powers, but again, the base must be the same. So for this one, we do the numbers first and subtract the indices second. This is because if I wrote the equation out fully, it would be the same as writing this. Just as you do when you simplify fractions, we can cancel down our numbers and our letters. So we cancel the 20 and the 5 to become 4, and we start by canceling out the b's. So to divide indices, we divide the numbers and we subtract the powers. Where do you want to go next? So what do you notice about these three? When powers are raised to a power, we multiply the powers together. On this one, be a little careful, don't forget because the three is also inside the brackets, it needs squaring two, hence nine, but then six multiplied by two is twelve. If we wrote this out, it would be 3B6 multiplied by 3B6, and we know that when we multiply indices, we add the powers, so 3 multiplied by 3 is 9 and 6 plus 6 is 12. So just remember, when we raise a power to a power, we just multiply the two powers together. Where do you want to go next? So what do you notice about the power of zero? Anything to the power of zero is one, but why is this? Let's use our knowledge of fractions and also dividing indices to see why. Do you agree that anything divided by itself is one? So a to the power of three goes into a to the power of three once. That's our first equation. And then using our indices knowledge, when we divide, we subtract the powers, so three minus three is zero, and that's our second equation. We've done the same calculation in two different ways. Therefore, the two answers must equal one another. So combining these two things, anything to the power of zero equals one. So where do you want to go next? Here are some questions for you to give a go yourself. Pause the video, work out the answers, and click play when you're ready to check. Did you get them right? So here are the four laws that we've seen so far. Make sure you understand why they work, and then you don't need to stress about learning them. Watch part two to discover how fractional and negative indices work.