x^(1/3) ... is it the same as x^(2/6) ?
lets consider x < 0
x^(1/3) is okay
(-8)^(1/3) = -2
then
x^(1/3) = (x^2)^(1/6) is okay since (-8)^2 = 64^(1/6) ((but this = 2))
and finally
x^(1/3) = (x^(1/6)^2) is not okay since (-8)^(1/6) = undefined
does anyone have an explanation, idea or comment about this? my math teacher brought this up to me today
Well, lets see, we have:
And we want to know if:
is the same.
Well, let's use the root versions of these:
and
Ok, let's use 8:
Well, we are going to assume that it is 2, well, what's 2 to the 6th power?
Looks like they're the same to me, now let's try -8
Looks like they're only the same for all .
What I mean by funky is that the two functions have opposite signs once your domain goes into the negative integers.
And the bottom just says that x to some power n is equal to x to the power of (2/2)n for all x in the natural number line (0,1,2,3,...).
As we saw, when and and , then:
for all x > 0.