then the value of m is

Concept

For a fixed base a with a > 0 and a ≠ 1, the exponential function is one-to-one, so two powers of the same base are equal exactly when their exponents are equal: ap = aq implies p = q. Combined with the power-of-a-power rule (ap)q = apq, this turns an equation between nested powers of a common base into a plain algebraic equation in the exponents alone.

Applying it here

1. The given equation is a^(m^n) = (a^m)^n.

2. Rewrite the right side using the power-of-a-power rule: (am)n = amn.

3. Both sides now have the same base a, so equate the exponents: mn = mn.

4. Divide both sides by m (m ≠ 0) to isolate the exponent on m: mn − 1 = n. (m = 0 also satisfies the original equation trivially, but it does not fix m in terms of n and is not among the offered values, so it is set aside.)

5. Take the (n − 1)-th root of both sides to solve for m (this requires n ≠ 1, so the exponent 1/(n − 1) is defined, and takes the principal positive real root, the standard convention for this identity): m = n1/(n − 1).

Cross-check

Raise this value back to the power (n − 1): (n1/(n − 1))n − 1 = n(1/(n − 1))·(n − 1) = n1 = n, which reproduces mn − 1 = n exactly — confirming the result.

So m = n1/(n − 1).