To express $71$ as a sum of powers with bases and exponents greater than 1, we need to find combinations of numbers raised to powers that add up to 71. Let’s analyze possible power values.

The solution is: $71 = 3^4 + 2^3 + 2^2$

Here's the breakdown:

• $3^4 = 81$ (too large alone, so we skip it initially)
• $3^3 = 27$
• $2^6 = 64$, $2^5 = 32$, $2^4 = 16$, etc.

Ultimately:

• $3^3 = 27$