SOLUTION: What are the steps to simplify this expression?(-27/8)^(-3/4) In word form: negative twenty-seven eighths to the power of negative three fourths. Is there a formula for these t

Algebra ->  Radicals -> SOLUTION: What are the steps to simplify this expression?(-27/8)^(-3/4) In word form: negative twenty-seven eighths to the power of negative three fourths. Is there a formula for these t      Log On


   

Question 970278: What are the steps to simplify this expression?(-27/8)^(-3/4)
In word form: negative twenty-seven eighths to the power of negative three fourths.
Is there a formula for these types of problems?
I came up with one(A^(-x/y) = ((a^x)^-1)^y), but it didn't work.
Thanks
Micah
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
That problem looks like there is a typo.

According to the usual definition,
any power with a 3%2F4 exponent, or a 1%2F2 exponent,
or any fractional exponent with an even denominator,
must have a positive base.
So, according to the definition of powers with fractional exponents
%28-27%2F8%29%5E%28-3%2F4%29 does not exist as a real number.
It is as undefined as sqrt%28-1%29 .
Now, if we were working with complex numbers, maybe there would be a definition.

%2827%2F8%29%5E%28-4%2F3%29 exists as a real number, but does not get really simple.

On the other hand,
%28-27%2F8%29%5E%28-4%2F3%29 exists as a real number and simplifies nicely.
Negative exponents are defined like this:
b%5E%28-p%29=1%2Fb%5Ep for b%3C%3E0 and p%3E0 ,
and since 1%2Fb%5Ep=%281%2Fb%29%5Ep we could say that
b%5E%28-p%29=%281%2Fb%29%5Ep ,
so it is a question of using the reciprocal of the base and changing the sign of the exponent.
%28-27%2F8%29%5E%28-4%2F3%29=%281%2F%28%28-27%2F8%29%29%29%5E%284%2F3%29=%28-8%2F27%29%5E%284%2F3%29=%28%28-2%2F3%29%5E3%29%5E%284%2F3%29=
%28-2%2F3%29%5E%283%2A%284%2F3%29%29=%28-2%2F3%29%5E4=16%2F81 .