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⚡ AZUCATION SHORTCUT:
Convert to powers of 10.
$(5^4)^{65} \times (2^7)^{36} = 5^{260} \times 2^{252} = 5^8 \times (10)^{252}$.
The digits are from $5^8 = 390625$, followed by zeros. Sum = $3+9+0+6+2+5 = \mathbf{25}$.

Step 1: Express bases as powers of prime numbers
$625 = 5^4$
$128 = 2^7$

Step 2: Substitute and simplify the expression
Expression $= (5^4)^{65} \times (2^7)^{36}$
$= 5^{4 \times 65} \times 2^{7 \times 36}$
$= 5^{260} \times 2^{252}$

Step 3: Factor out the powers of 10
To find the digits, we isolate $10^n$ (which is $2^n \times 5^n$):
$= 5^8 \times 5^{252} \times 2^{252}$
$= 5^8 \times (5 \times 2)^{252}$
$= 5^8 \times 10^{252}$

Step 4: Calculate the non-zero part
$5^8 = (5^4)^2 = 625^2 = 390625$.
The full number is $390625$ followed by $252$ zeros.

Step 5: Final Sum of Digits
Sum $= 3 + 9 + 0 + 6 + 2 + 5 + (252 \times 0)$
Sum $= \mathbf{25}$.

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