2.4 Multiplying and dividing powers

00:00:05 in this video we're going to look at questions like these two so ask how to multiply or divide numbers when they have powers the first thing to say though is that the rules that we're going to cover only work if the numbers involved both have the same base for example in this multiplication one they both have a base of b and in this division one they both have a base of a whereas if we had a question like b cubed times a c squared we wouldn't be able to multiply them together because the bases of b and c

00:00:42 are different letters the easiest way to understand this topic is to think about what the powers really mean if we start with this multiplication one b cubed just means b times b times b because the power of three means that we have three b's being multiplied together meanwhile b squared just means b times b and so if we multiply these two terms together we're just gonna have five b's being multiplied together which is the same thing as b to the power of five so that's our answer instead of writing all of this out every

00:01:24 time though the shortcut is to just add together the two powers so add together the three and the two to get five as our new power and we can show the working out for this as b cubed times b squared equals b to the power of 3 plus 2 which equals b to the power of 5. it works in a pretty similar way for division a to the power of 6 divided by a to the power of three really just means a times a times a times a times a times a so six a's multiplied together divided by a times a times a which is three a's multiplied together

00:02:10 and then to simplify that we could just cancel out a's on the top and the bottom until we find that would only have three a's left on the top and that tells us that our answer is a cubed again though there is a quicker way to do this which when we're dividing is to subtract the two powers so for a to the power of six divided by a to the power of three we just write a to the power of six minus three which equals a cubed let's try a couple of questions now so in this first one we're doing x to the power of seven times x to the power

00:02:53 of four so because we're multiplying we need to add the powers so x to the power of seven plus four which is x to the power of 11. this next one 2 to the power of 7 divided by 2 to the power of 4 is a bit different because the base is a number rather than a letter but it works in exactly the same way because we're dividing this time we have to subtract the powers so two to the power of seven minus four which is two to the power of three one extra thing we can do when the base is a number though is to work out its

00:03:34 exact value so because 2 to the power of 3 is just 2 times 2 times 2. we can multiply them all together to find the answer is 8. for this next one a to the power of nine times a to the power of minus five we again just add the powers so a to the power of nine plus negative five which is the same thing as a to the power of nine minus five so a to the power of four then for b cubed divided by b to the power of eight we just do b to the power of three minus eight which is b to the power of negative five

00:04:21 and it doesn't matter that our power here is negative that's absolutely fine in this next one we have three terms being multiplied together but the technique that we use is exactly the same we just add together all of the powers so a to the power of two plus four minus three which overall gives us a to the power of three for p cubed divided by p the p term looks like it doesn't have a power but whenever you have a number or letter by itself it really just has an unwritten power of one so we just do p to the power of three

00:05:06 minus one which is p squared one last thing to cover is that when you're doing questions like these that involve multiplying or dividing terms that have both numbers and letters you want to think of the numbers and letters separately and remember that here the powers only apply to the letters so for three a to the power of five times four a cubed you would do three times 4 which is 12 for our numbers and times that by a to the power of 5 plus 3. so a to the power of 8. and that gives us an overall answer of

00:05:51 12 a to the 8th or for 8 b to the power of 11 divided by 2 b to the power of 5 you have to do 8 divided by 2 which is 4 times b to the power of 11 minus five which is b to the power of six so your answer would be four b to the power of six anyway that's everything for this video so hope you found it useful if you did then please do give us a like and subscribe and hopefully we'll see you again soon