Find the total number of prime factors in the expression 411×75×112

When a number is written as a product of prime powers p1a×p2b×..., the total number of prime factors counted with multiplicity (repetition) is simply the sum of the exponents a+b+... — each exponent tells you how many times that prime is repeated in the product.

1. 4 is not itself a prime number, so first rewrite it in terms of its prime base: 4 = 22.

2. Apply the power-of-a-power rule (multiply the exponents, not add them): 411 = (22)11 = 22×11 = 222, so this term contributes 22 factors of 2.

3. 75 is already prime, contributing 5 factors of 7.

4. 112 is already prime, contributing 2 factors of 11.

5. Add the contributions from every prime term: 22 + 5 + 2 = 29.

Cross-check: rewriting the whole expression purely in prime bases gives 222×75×112 — counting the exponents once more (22, 5, 2) and re-adding them confirms the same total of 29 prime factors.