Evaluate 4^2^2^2 - 1.

Step-by-step solution

1. 1

   Fix the direction of evaluation first — it changes the answer. Stacked exponents are right-associative: $a^{b^{c}}$ means $a^{(b^{c})}$, never $(a^{b})^{c}$. Reading left to right here would give $((4^{2})^{2})^{2}=4^{8}=65536$, which is wrong by five orders of magnitude. So work from the top down.

2. 2

   Collapse the top of the tower.

   $2^{2}=4,\qquad\text{then}\qquad 2^{2^{2}}=2^{4}=16$

   The exponent of the base $4$ is therefore $16$, and the expression becomes $4^{16}-1$.

3. 3

   Rewrite the base as a power of 2 to keep the numbers manageable. Since $4=2^{2}$, the power-of-a-power rule multiplies exponents:

   $4^{16}=\left(2^{2}\right)^{16}=2^{32}=4\,294\,967\,296$

   Recognising $2^{32}$ is the shortcut — it is a familiar constant, so no long multiplication is needed.

4. 4

   Subtract 1. The subtraction applies to the whole power, not to the exponent:

   $4^{16}-1=4\,294\,967\,296-1=4\,294\,967\,295$

5. 5

   Verify exactly. Python's arbitrary-precision integers give $4^{(2^{(2^{2})})}-1=4294967295$ ✓ and $2^{32}=4294967296$ ✓. As a structural check, $2^{32}-1$ is the largest unsigned 32-bit integer, and it factors as $(2^{16}-1)(2^{16}+1)=65535\times65537$ — a difference of squares once more.