(1) If 5^m=4^m 2ⁿ, then 5 = 2^{2m + n} which is impossible as 2 and 5 are prime so there are no positive integer solutions m and n of this equation.

(2) We have a^m dⁿ=cⁿ b^m. As a and b are coprime, we get a^m|cⁿ. Because c and d are coprime, so cⁿ|a^m. This means a^m=cⁿ. Similarly, b^m=dⁿ.

If a^m=cⁿ and b^m=dⁿ, then obviously (a/b)^m=(c/d)ⁿ.

This implies that a^m and cⁿ have the same prime factors. Write a = p₁^u₁...p_k^u_k and c = p₁^v₁...p_k^v_k and for all j we have mu_j = nv_j so that

u1 v1 = u2 v2 = ... = uk vk = m n .

Similarly for b and d. This is a very special necessary relationship between a and c and also between b and d so solutions rarely occur to the equation:

æ ç è 5 4 ö ÷ ø m   = æ ç è 2 1 ö ÷ ø n   .

We now show this is a sufficient condition. Conversely suppose a = p₁^u₁...p_k^u_k and c = p₁^v₁...p_k^v_k and

u1 v1 = u2 v2 = ... = uk vk .

We call this common ratio ⁿ/_m then u_jm=v_jn for all j and a^m = bⁿ. Similarly if corresponding ratios of the powers of the prime factors of b and d are constant and also equal to ⁿ/_m then b^m=dⁿ giving

æ ç è a b ö ÷ ø m   = æ ç è c d ö ÷ ø n   .