First of all. Negative exponents. A negative exponent means take the reciprocal.
a^(-n) = 1/(a^n)
(1/b)^(-n) = b^n
(a/b)^(-n) = (b/a)^n
Now, fractional exponents. A fraction exponent indicates a root.
a^(1/2) = sqrt(a)
a^(1/3) = cube root of a
a^(1/4) = fourth root of a
a^(3/5) = fifth root of (a^3) = [fifth root of a]^3
Other exponent rules:
(a^m)*(a^n) = a^(m + n)
(a^m)/(a^n) = a^(m - n)
[a^m]^n = a^(m*n)
Now, to your questions:
1) (1/4)^-1/4 = 4^(1/4) = [4^(1/2)]^(1/2) = 2^(1/2) = sqrt(2)
2) (4)^-1/2 = (1/4)^(1/2) = 1/2
3) (1/4)^1/4 = [(1/4)^(1/2)]^(1/2) = (1/2)^(1/2) = 1/sqrt(2)
Does this make sense? Here's a free lesson video on exponents:
https://gmat.magoosh.com/lessons/216-intro-to-exponents
Just so you know, at Magoosh we have 200+ lesson videos and 800+ practice questions, each with its own video explanation --- all for a fraction of the price of other test prep sources. If you have lots of questions on basic math, Magoosh video lessons might be the key for you.
Let me know if you have any questions.
Mike
