Find the total number of prime factors in the expression $(4)^{11} \times (7)^5 \times (11)^2$.

### Concept & Logic Prime factorization requires converting all composite bases in an expression into their prime factors before summing the exponents. ### Step-by-Step Solution **Given:** Expression: $(4)^{11} \times (7)^5 \times (11)^2$. **Calculation / Deduction:** * Identify the prime and composite bases. Here, $7$ and $11$ are already prime numbers, but $4$ is composite. * Convert $4$ to its prime base: $4 = 2^2$. * Substitute this back into the expression: $$ (2^2)^{11} \times (7)^5 \times (11)^2 $$ * Apply the exponent rule $(a^m)^n = a^{m \times n}$: $$ 2^{22} \times 7^5 \times 11^2 $$ * The total number of prime factors is simply the sum of these prime exponents: $22 + 5 + 2 = 29$. ### Exam Strategy & Shortcut To find total prime factors quickly, just multiply the exponent of any composite number by the count of its prime components. Since $4 = 2 \times 2$ (two components), its exponent contribution is $11 \times 2 = 22$. Then just add the rest: $22 + 5 + 2 = 29$. ### Common Pitfall The most common mistake is ignoring the composite base and directly adding the given exponents ($11 + 5 + 2 = 18$). Always verify that every base is a prime number before summing. ### Final Answer Therefore, the correct answer is 29.