What is the number of prime factors contained in the product $30^7 \times 22^5 \times 34^{11}$?

### Concept & Strategy To count the total prime factors in an exponential product, you must break down every composite base into its prime building blocks. Apply the exponent to each individual prime, and then sum all the resulting exponents. ### Step-by-Step Solution * **Given:** * The product expression: $30^7 \times 22^5 \times 34^{11}$ * **Calculation / Deduction:** 1. Prime factorize each composite base independently: $30 = 2 \times 3 \times 5$ (3 prime factors) $22 = 2 \times 11$ (2 prime factors) $34 = 2 \times 17$ (2 prime factors) 2. Apply the original exponents to these prime factorizations using the rule $(abc)^n = a^n \cdot b^n \cdot c^n$: $30^7 = (2 \times 3 \times 5)^7 = 2^7 \times 3^7 \times 5^7$ $22^5 = (2 \times 11)^5 = 2^5 \times 11^5$ $34^{11} = (2 \times 17)^{11} = 2^{11} \times 17^{11}$ 3. Count the total prime factors by summing all the new exponents: From $30^7$: $7 + 7 + 7 = 21$ factors From $22^5$: $5 + 5 = 10$ factors From $34^{11}$: $11 + 11 = 22$ factors 4. Calculate the final sum: $$Total = 21 + 10 + 22 = 53$$ ### Exam Strategy & Shortcut Count the number of prime components in each base and multiply by the exponent. $30$ has $3$ components ($2,3,5$): $3 \times 7 = 21$. $22$ has $2$ components ($2,11$): $2 \times 5 = 10$. $34$ has $2$ components ($2,17$): $2 \times 11 = 22$. Sum $= 21 + 10 + 22 = 53$. This method skips rewriting the entire algebraic string. ### Common Pitfall A common error is to just prime factorize the base and forget to distribute the exponent to *every* prime factor. For instance, treating $30^7$ as just $7$ prime factors instead of $21$. ### Final Answer Therefore, the correct answer is 53.