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We're asked to simplify the cube root of twenty seven a squared times b to the fifth
times c to the third power.
And the goal whenever you try to just simplify a cube,a cube root like this, is we wanna look
at the parts of this exepression
over here,
that are perfect cubes that are something raised to the 3rd power
that we can take just the cube root of those, since you taking them out of the radical sign
and then leaving everything else that is not a perfect cube inside of it
so let's see what we can do
so first of all 27, you may or may not already recognise this
is a perfect cube,if you don't already recognise it, you can
actually do a prime factorisation and see it is a perfect cube
27 is 3 times 9 and
9 is 3 times 3 so 27
prime factorisation is 3 times 3 times 3, so is the exact
same thing as 3 to the third power, so
let's rewrite this whole expression down here, let's try in terms of things
that are perfect cube and things that aren't, so 27 can be just rewritten
as 3 to the third power, then you have
a squared, clearly not a perfect cube, but if the third would have been
so we just gonna write this, let me write it over here
we can switch the order here, cause we just have a bunch of things being multiple by each other, so i'll write
the a squared over here,b to the fifth
b to the fifth is not a perfect cube by itself
it can be, it can be expressed as a product of a perfect cube
in another thing, b to the fifth is the exact same thing as the
b to the third power times b to the second power
if you wanna see that explicitly b to the fifth is b times b times
b times b times b
so this, the first three are clearly b to the third power
and then you have b to the second power after it, so we can rewrite
b to the fifth as a product of a perfect cube, so
i'll write b to the third (doing the same purple colour)
so we have b to the third power over here
and this b to the thirds times b squared, so i'll write the b squared
over here, we assume we gonna multiply all this stuff, and then finally
finally, we have(other than blue)
c to the third power, clearly this is a perfect cube
it is c cube,it is c to the third power, so i'll
put it over here, so this is c to the third power, and of cause we still have
that over arching radical sign, so we still
trying to take the cube root of all of this, and we know
from our exponent properties, or we could say from our radical properties
that this is the exact same thing, that taking the cube root of all of this
things is the same as taking the cube root of these individual factors
and then multiplying them, so this is the same thing as
the cube root, and I can separate them out individually or I could say that
cube root of 3 to the third, b to the third
c to the third,I'll have to use it both ways, so i'll start taking them
out separately, so this is the same thing as the cube root of
three to the third times the cube root ( I'll write
them all it, while I write do it we'll colour code so we all don't get confused )
times the cube root of to b to the third times
the cube root times the
cube root of c to the third
c to the third, times the cube root
and i'll just group these two guys together just because we're not gonna be able to
simply them anymore, times the cube root, times the
cube root of a squared b squared
( I'll keep the colour consistent while we are trying to figure out what's what )
a squared, b squared, now I could have, could have said
that this is times the cube root of a squared times the cube root of b squared, but that won't
simplify anything, but that won't simplify anything so i'll just leave these like this, and so we
can look as these individually the cube root of 3 to the third or the cube root of
27, well that's clearly just gonna be ( i'm gonna do that in that
yellow colour ) this is clearly just gonna be 3
right, 3 to the third power is three to the third power
or it is equal to 27, this term right over here, the cube root
of b to the third, well that is just b, that's just b
and the cube root of c to the third, the cube root of c to the third, well
that is clearly (won't do that in that ) that is
clearly just c
So our whole expression have simplified to
3 times b times c
times c, times the cube root
times the cube root, of a squared b squared
times the cube root of a squared
a squared, b squared, and we're done
i just wanna do just one other thing, just cause I did mentioned I would do it
we could simplify it this way, or we could recoignise, we could recoignise
that this expression right over here can be written as
3 b c
to the third power, but if I take 3 things to the third power and i'm multiplying it, that is the same thing as
multiplying them first and then raising to the third power, comes straight out
of our exponent properties, and so we can rewrite this as
cube root, the cube root of all of this times
the cube root, times the cube root of a squared
b squared, and so the cube root of all this of 3 b
c of the third power, well that's just gonna be 3
b c and then multiply it by the cube root of
a squared b squared, I didn't take the trouble to colour code it this time cause we already
figured out one way to solve it, but hopefully that also makes sense, we could have gone
that we could have done this either way, but the important thing is that we get the same answer.