Simplify (1/4)^-1/4 (show work).

Hello and thank you for your question.

I reach the same answer as the comment below, but maybe in an easier way.

Since the negative exponent means "one over," you have
1 / ((1/4)^1/4)

That denominator, ((1/4)^1/4) , can be rewritten as 1 / (4)^1/4

So the entire formula,
1 / ((1/4)^1/4)

is the same as
(4)^1/4

in other words just as
  1 / (1 / A)  = A
 
  1 / ((1 / (4)^1/4)  = (4)^1/4

and since 2^2 = 4

(4)^1/4 = (2^2)^1/4

or

2^(2/4) = 2 ^(1/2) = = sqrt(2)

Thanks again for letting us help.

Sincerely,
Google Answers Researcher
Richard-ga

I assume you mean (1/4)^(-1/4).

This problem may look tricky, but once you break it down, it's rather
simple.  Here's how to go about it:

First, remember that by making the exponent negative, you can 'flip'
the fraction over.  So 1/4 equals 4^(-1).  Then you can rewrite your
problem as (4^(-1))^(-1/4).  Then, remember that that 4=2^2, and
rewrite it like this: ((2^2)^(-1))^(-1/4).

Now, remember that (a^b)^c = a^(bc).  2*-1 is equal to -2, and
-2*(-1/4) equals 1/2, so you can rewrite the expression again as
2^(1/2)

And guess what? 2^(1/2) is just the square root of two!

So, (1/4)^(-1/4) = sqrt(2).  yay!

hope that helps :-)