Question

3. For any positive integer m and n, and any non-zero rational number a, we define a to the power m n = n√a m and a to the power m n = n√a -m = n√a-m = 1 by n√a m. ​

Answer 1

Answer:

If n is a positive integer then m^n will be an integer for any value of m (taking into account that both are nonzero integers).

If n is negative then m^n will be an integer if and only m=1 or m=-1, for example: (-1)^(-2)=1/(-1)^2=1

So basically we are asked: is n positive or m=|1|?

(1) n^m is positive --> either m=even (and in this case n can take any value) or n=positive (and in this case m can take any value). Not sufficient.

(2) n^m is an integer --> either m=positive (and in this case n can take any value) or m=negative and in this case n=1 or -1. Not sufficient.

(1)+(2) If n^m=(-1)^2=positive integer, then the answer will be NO as m^n=2^(-1)=1/2 but if n^m=1^2=positive integer, then the answer will be YES as m^n=2^1=2. Not Sufficient.