Final answer:
The expressions equivalent to 3^12 • 7^9 are option C) 7^3 • (3^-4)^-3 • 7^6 and option E) 3^20 • (7^3)^3 • (3^4)^-2, utilizing the rules of exponents.
Step-by-step explanation:
We are tasked with finding which expression is equivalent to 3^12 • 7^9. Let's analyze the given options one by one:
- A) 3^3 • 3^4 • 4^9: This option cannot be correct because it introduces a base of 4, which does not appear in the original expression.
- B) (3^3)^9 • (7^3)^6: Applying the rule of exponents, when you raise a power to a power, you multiply the exponents. Hence, (3^3)^9 becomes 3^(3×9)=3^27, and (7^3)^6 becomes 7^(3×6)=7^18, which is not the original expression 3^12 • 7^9.
- C) 7^3 • (3^-4)^-3 • 7^6: Again using the rule of exponents, (3^-4)^-3 becomes 3^12 and 7^3 • 7^6 simplifies to 7^9. This matches the original expression 3^12 • 7^9, so this option is correct.
- D) (3^3 + 3^9) • (7^6 + 7^3): This option introduces addition and doesn't match the multiplication of exponents in the original expression.
- E) 3^20 • (7^3)^3 • (3^4)^-2: This expression simplifies to 3^20 • 7^9 • 3^-8, which eventually simplifies to 3^12 • 7^9. Thus, this option is also correct.
Hence, the equivalent expressions to 3^12 • 7^9 are C) 7^3 • (3^-4)^-3 • 7^6 and E) 3^20 • (7^3)^3 • (3^4)^-2.