In this video we’re going to look at how we can work with indices or powers, those are those little numbers that sits at up in the air on other numbers. So we’re going to start off with an example of looking at 3 to the power of 2 or 3 squared times by 3 to the power of 3 or 3 cubed 3 to the power of 2 means? 3 is written down twice and times together. 3 to the power of 3 means, 3 is written down 3 times and times together and in this sum, we’re timesing those two bits together, so we’ve got 3 written down 5 times and times together, so that could be written! It’s 3 to the power of 5.
Now what you can actually do with these, you can just spot that these powers here these indices, just get added together. So the 2 and the 3 added together to give us the 5. So if you’re, multiplying and the big numbers here are the same with each other and that’s important, these big numbers have to be the same with each other. If you’re multiplying, you can just add the powers together. So if you have a look at another example, if you have a look at 7 to the power of 5 times by 7 to the power of 6, because the big numbers are the same with each other and we’re timesing, that’s the same as 7. The bigger mistake, the same to the power well, 5, add 6 is 11, so that would be the same as 7 to the power of 11. We can use these rules when we’re dividing as well! If we had 5 to the power of 8 divided by 5 to the power of 6, you won’t be surprised. The earth´s has got a big 5 in it as well, but when we were timesing, we added the powers.
If you’re dividing you do a taking away with the powers and 8 take away! 6 is 2, so that one would be the same as 5 to the power of 2. If you’re dividing you do a taking away with the powers, so look at another one? If we had 3 to the power of 4 divided by 3 to the of five, the answer will have a big three, because these numbers are the same they’re, both threes and if we’re dividing, we can do it taking away so i’m going to do four. Take away! Five and i’ll just write that working out down four take away! Five now four take away. Five is going to take you into negative numbers, because you can’t take five away from four! There isn’t enough there for take away. 5 is minus one, so you can get answers whether a minus powers. It’s also worth mentioning at this point, 3 to the power of minus 1. Any number to the power of minus 1 is the same as 1 over that number. So 3 to the power of minus 1 is the same as 1 over 3 or 1/3. If i’ve got 7 to the power of minus 1 from another question, that would be the same as 1 over 7.
Moving on you can have other questions which give you num – purse as well. We look at 4 to the power of 4, divided by 4 to the power of 7. We’ve got to do a taking away because we’re dividing 4 take away. 7 is minus 3, so we have 4 to the power of minus 3?
The minus power means 1 over just like it did here, but it’s not just 1 over 4, because that would be 4 to the minus 1. If there’s a three here, that’s got to come down here as well, so 4 to the minus 3 means 1 over 4 to the power of 3. Remember 4, to the power of 3 means 4 times 4 times 4. If you worked out 4 times 4 times 4 for 4 to 16 times that by 4 you get 64, so that is actually the same as 1 over 64. If we look at another one 7 squared times by 7 cubed over 7 to the power of 4, remember and over like this means a divide where to the top first on the top of the fraction, which is the numerator of the french, we’re timesing. So we can add the little numbers! Add the pairs together. Two plus three is five, so that’s the same as 7 ^ 5 on the bottom i’ve got a 7 ^ 4? Now i said that an over like this align here with something’s over something means the divide.
So that question is exactly the same as 7 ^ 5/7 ^ 4, because we’re dividing we can do a taking away 5 take away? 4 is 1!
That’s 7, ^ 1, it’s worth remembering that anything to the power of 1 is just a number itself, so 7 ^ 1 is just 7. If i’d had another question and then it’d give me an answer of 12 to the power of 1, that would just be the same as 12. Now we can use these laws of indices as they’re called in algebra, as well as with numbers, if i had x, squared times by x cubed. That’s really quite important. Now that you make your x’s and your time signs look different! That’s one reason why we use these curvy x’s, because i’m timesing- and these are the same thing here- they’re both x’s i’ll- have that excellent answer in to add 3 is 5 so 4 that will not have that’s the same as x to the power of 5.
If i had another example, 2x squared times by 4 x cubed, here i’ve got a mixture of big numbers, the 2 and the 4 and powers, the 2 and the 3. You deal with the big numbers, normal numbers first 2 times, 4 is 8, so effectively know that – and that 4 have gone and you’re left with an x squared times by an x cubed, so that’ll be an x and when you add their powers, just like in the last question, you’ll have x to the power of 5! So the final answer there will be 8 x to the power of 5.
We have a look at another one like that: 3 y ^ 7 x by 4 y ^ 3 deal the big numbers first, three times by 4 equals 12, and then you can use the laws of indices on the y’s y to the power 7 times by y to the power of 3, because we’re timesing, we can add the powers and we’ll get y to the power of 10!
We can do this for dividing as well if we had a to the power of 7 divided by a to the power of 4, because that’s what this means here if you’ve got an over as a divided, that’s just the same as a to the power of 7, divided by a to the power of 4. Remember when you’re dividing you do taking away. So that would be the same as a to the power of 3? If we look at some other examples, if i’ve had y to the power of 5 over y to the power of 8, that over means i’m dividing so i could write it down as y to the power of 5, divided by y to the power of 8. If i’m, dividing i can do taking away 5 take away, 8 is minus 3, so you can get these – powers coming up in the algebra examples as well, and the minus power just means 1 over what we had here without the minus sign.
So if i took the minus sign out, i’d have y cubed, so y to the minus 3 is the same as 1 over y cubed. If we look at just two more examples, probably three x to the power of seven times by four x to the power of six over six x squared make sure we can see that one. Let’s do all the top first and we’ll leave the bottom alone we’ll deal with that. One in a minute three times by four, is 12 x to the power of seven times x to the power of six!
To the add those powers will give me x to the power of 13. Now the question they’ve got is the same as x to the power of 13, divided by 6x squared deal with the normal numbers! First, two big numbers, 12 divided by 6, is 2 x to the power of 13, divided by x, to the power of 2, from dividing i, do taking away with the powers, so i’ll get to x to the power of 11 and the final one! For now, if you had a cubed, which was then all squared, that means a cubed x by itself a cubed x by a cubed. If you’re timesing, you add the powers, you can miss that middle step out a power to a power. You can just multiply those powers together.
Three times by two is six. You can get straight to that. So if you had x to the 7 to the power of 4, you could say that’s the same as x to the power of 28. !