If a is congruent to b(mod m) and c is congruent to d(mod m) then a^c is congruent to b^d(mod m)

Question

Prove or disprove:
If $a \equiv b \pmod m$ and $c \equiv d \pmod m$ then $a^c \equiv b^d \pmod m$.

Please help I've been working on this for days

Answer 1

Have you tried any examples?

How about $m=4$, $a=b=2$, and $c=1$, $d=5$.

Then $a \equiv b \pmod m$ and $c \equiv d \pmod m$.

But $a^c = 2^1 \equiv 2 \pmod 4$ and $b^d = 2^5 = 32 \equiv 0 \pmod 4$.

Thus $a^c \equiv 2 \not \equiv 0 \equiv b^d \pmod 4$.