0:07 in this video we're looking at how to
0:09 use fractional powers
0:11 like in the case of nine to the power of
0:13 three over two
0:16 where the power here is a fraction
0:18 these ones can be a bit confusing at
0:20 first so just bear with us through this
0:22 first example and then we'll go through
0:24 a whole bunch of them
0:26 whenever you see a fractional power like
0:29 this you want to think of the numerator
0:32 and the denominator separately
0:35 where the numerator tells you the power
0:37 that you need to raise your number to
0:40 and the denominator tells you the route
0:42 that you need to take
0:44 so here the 3
0:47 tells us that we need to cube the 9
0:49 and the 2 tells us that we need to take
0:51 the second root
0:53 which just means the square root
0:55 the problem is that we can't do both of
0:58 these operations at once so we're going
1:00 to have to do one of them first and then
1:03 the other one
1:06 for example we could do 9 cubed
1:09 which is 729
1:12 and then square root it to get 27
1:14 or we could take the square root of 9
1:16 first to get 3
1:20 and then cube that to get 27
1:22 and in general this second option where
1:24 we take the root first will be much
1:26 easier because we're working with
1:35 so overall the way i would do this question
1:36 question
1:38 is that when i see 9 to the power of 3
1:42 over 2 i'd rewrite it as the square root
1:43 of 9
1:45 all cubed
1:47 so that i could simplify the root 9 to a 3
1:48 3
1:51 so that i'm left with 3 cubed
1:53 and then i would just do 3 cubed to get 27
1:59 let's start off with some easier
2:02 questions where the fractional powers
2:04 all have a 1 on the top
2:07 like one half one third one quarter and
2:09 so on
2:12 because the numerators are all ones
2:14 we effectively don't have to worry about
2:15 the powers
2:18 we just take the root of whatever number
2:20 is on the bottom
2:23 so for 16 to the power of one half
2:26 we'd have the square root of 16
2:28 which is four
2:31 then for 27 to the power of a third
2:34 we'd have the third root of 27
2:37 which is 3
2:39 then 81 to the power of a quarter would
2:42 be the fourth root of 81
2:44 which is also 3
2:46 and the x to the power of one fifth
2:49 would be the fifth root of x
2:51 which we can't simplify any further so
2:58 now let's have a go at some slightly
2:59 harder ones
3:02 for this first one we're trying to do 8
3:05 to the power of two thirds
3:08 so because this three means cube root
3:10 and is two means square
3:13 you want to rewrite it as a cube root of eight
3:14 eight
3:16 all squared
3:18 which we can then simplify
3:21 so the cube root of eight is two
3:24 and two squared is four
3:26 so our answer's four
3:30 next up we have 27 to the power of five thirds
3:31 thirds
3:32 which means that we're going to have the
3:34 cube root of 27
3:37 all to the power of 5
3:39 which we can then simplify to 3 to the
3:44 power of 5 which is 243
3:46 this last one is a bit trickier because
3:48 our fractional power this time is negative
3:50 negative
3:51 so we're gonna have to sort that out first
3:53 first
3:55 if you remember from our previous video
3:57 whenever you have a negative power all
3:59 you need to do is flip the whole thing
4:00 upside down
4:03 turning it into 1 over 16 to the power
4:05 of negative 3 over 2
4:08 and then make the power positive
4:10 so change the minus 3 over 2 to positive
4:13 3 over 2.
4:15 so basically this means exactly the same
4:18 thing but it now has a positive power
4:20 which makes it easier to work with
4:21 meaning that we can use the same
4:25 technique as we were using before
4:28 so we do 1 over the square root of 16 cubed
4:29 cubed
4:33 which simplifies to 1 over 4 cubed or
4:40 the last thing we need to look at is
4:42 what happens when you have a fraction
4:46 that's raised to a fractional power
4:48 these questions basically use all of the
4:50 rules that we've learned so far in one go
4:52 go
4:54 so for nine over sixteen to the power of
4:55 three over 2.
4:58 the first thing we'd do is apply the 3
5:00 over 2 power to the numerator and
5:03 denominator separately
5:07 so 9 to the power of 3 over 2
5:12 divided by 16 to the power of 3 over 2.
5:14 next we can rewrite the top as the
5:17 square root of 9 cubed
5:20 and the bottom as the square root of 16 cubed
5:22 cubed
5:24 which simplifies to 3 cubed
5:25 cubed
5:27 over 4 cubed
5:28 cubed
5:35 for this last one we do basically the
5:36 same thing
5:38 but because there's a negative sign on
5:40 our power this time
5:41 we're gonna have to flip the fraction
5:43 upside down first
5:47 so that it becomes 8 over 125
5:52 or to the power of positive 4 over 3.
5:54 next we're going to take our power and
5:56 apply it to the numerator and
5:58 denominator separately
6:02 so we have 8 to the power of 4 over 3
6:07 divided by 125 to the power of 4 over 3
6:09 and then we can rewrite that as the cube
6:12 root of 8 to the power of 4
6:15 over the cube root of 125 to the power
6:21 and then because the cube root of 8 is 2
6:25 and the cube root of 125 is 5
6:29 we can simplify it to 2 to the power 4
6:32 over 5 to the power of 4
6:42 anyway that's everything for this video
6:44 so i hope you found that useful if you
6:46 did then please do tell your friends and
6:48 teachers about us and cheers for watching
