The number of prime factors in the expression $6^{10} \cdot 7^{17} \cdot 11^{27}$ is equal to

### Concept & Strategy To find the total number of prime factors in a product of exponents, you must first prime factorize every base. Once all bases are prime, the total number of prime factors is simply the sum of all the exponents. ### Step-by-Step Solution * **Given:** * The mathematical expression: $6^{10} \cdot 7^{17} \cdot 11^{27}$ * **Calculation / Deduction:** 1. Inspect the bases: $6$, $7$, and $11$. 2. Base $7$ and base $11$ are already prime numbers. 3. Base $6$ is a composite number. Prime factorize it: $$6 = 2 \times 3$$ 4. Substitute this back into the expression using exponent rules $(a \cdot b)^n = a^n \cdot b^n$: $$6^{10} = (2 \times 3)^{10} = 2^{10} \cdot 3^{10}$$ 5. Rewrite the full expression with only prime bases: $$2^{10} \cdot 3^{10} \cdot 7^{17} \cdot 11^{27}$$ 6. The exponents represent how many times each prime factor appears. Add the exponents to find the total count: $$Total Prime Factors = 10 + 10 + 17 + 27 = 64$$ ### Exam Strategy & Shortcut Scan the bases. $7$ and $11$ are prime, contributing $17$ and $27$ directly. $6$ breaks into two primes ($2$ and $3$). Thus, the power of $10$ applies twice ($2 \times 10 = 20$). Add them up: $20 + 17 + 27 = 64$. You can do this entirely mentally. ### Common Pitfall The most frequent error is ignoring the base $6$ and simply adding the given exponents: $10 + 17 + 27 = 54$. Always ensure every single base is reduced to prime numbers before summing powers. ### Final Answer Therefore, the correct answer is 64.