This content explains how to calculate numbers raised to fractional powers by breaking down the numerator and denominator into separate operations: exponentiation and root extraction.
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in this video we're looking at how to
use fractional powers
like in the case of nine to the power of
three over two
where the power here is a fraction
these ones can be a bit confusing at
first so just bear with us through this
first example and then we'll go through
a whole bunch of them
whenever you see a fractional power like
this you want to think of the numerator
and the denominator separately
where the numerator tells you the power
that you need to raise your number to
and the denominator tells you the route
that you need to take
so here the 3
tells us that we need to cube the 9
and the 2 tells us that we need to take
the second root
which just means the square root
the problem is that we can't do both of
these operations at once so we're going
to have to do one of them first and then
the other one
for example we could do 9 cubed
which is 729
and then square root it to get 27
or we could take the square root of 9
first to get 3
and then cube that to get 27
and in general this second option where
we take the root first will be much
easier because we're working with
so overall the way i would do this question
question
is that when i see 9 to the power of 3
over 2 i'd rewrite it as the square root
of 9
all cubed
so that i could simplify the root 9 to a 3
3
so that i'm left with 3 cubed
and then i would just do 3 cubed to get 27
let's start off with some easier
questions where the fractional powers
all have a 1 on the top
like one half one third one quarter and
so on
because the numerators are all ones
we effectively don't have to worry about
the powers
we just take the root of whatever number
is on the bottom
so for 16 to the power of one half
we'd have the square root of 16
which is four
then for 27 to the power of a third
we'd have the third root of 27
which is 3
then 81 to the power of a quarter would
be the fourth root of 81
which is also 3
and the x to the power of one fifth
would be the fifth root of x
which we can't simplify any further so
now let's have a go at some slightly
harder ones
for this first one we're trying to do 8
to the power of two thirds
so because this three means cube root
and is two means square
you want to rewrite it as a cube root of eight
eight
all squared
which we can then simplify
so the cube root of eight is two
and two squared is four
so our answer's four
next up we have 27 to the power of five thirds
thirds
which means that we're going to have the
cube root of 27
all to the power of 5
which we can then simplify to 3 to the
power of 5 which is 243
this last one is a bit trickier because
our fractional power this time is negative
negative
so we're gonna have to sort that out first
first
if you remember from our previous video
whenever you have a negative power all
you need to do is flip the whole thing
upside down
turning it into 1 over 16 to the power
of negative 3 over 2
and then make the power positive
so change the minus 3 over 2 to positive
3 over 2.
so basically this means exactly the same
thing but it now has a positive power
which makes it easier to work with
meaning that we can use the same
technique as we were using before
so we do 1 over the square root of 16 cubed
cubed
which simplifies to 1 over 4 cubed or
the last thing we need to look at is
what happens when you have a fraction
that's raised to a fractional power
these questions basically use all of the
rules that we've learned so far in one go
go
so for nine over sixteen to the power of
three over 2.
the first thing we'd do is apply the 3
over 2 power to the numerator and
denominator separately
so 9 to the power of 3 over 2
divided by 16 to the power of 3 over 2.
next we can rewrite the top as the
square root of 9 cubed
and the bottom as the square root of 16 cubed
cubed
which simplifies to 3 cubed
cubed
over 4 cubed
cubed
for this last one we do basically the
same thing
but because there's a negative sign on
our power this time
we're gonna have to flip the fraction
upside down first
so that it becomes 8 over 125
or to the power of positive 4 over 3.
next we're going to take our power and
apply it to the numerator and
denominator separately
so we have 8 to the power of 4 over 3
divided by 125 to the power of 4 over 3
and then we can rewrite that as the cube
root of 8 to the power of 4
over the cube root of 125 to the power
and then because the cube root of 8 is 2
and the cube root of 125 is 5
we can simplify it to 2 to the power 4
over 5 to the power of 4
anyway that's everything for this video
so i hope you found that useful if you
did then please do tell your friends and
teachers about us and cheers for watching
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