Final answer:
The expression (3a)^3(4a^4)^2 expands and evaluates to 432a^11. We use the rules for cubing of exponentials and multiplying like bases, resulting in 27a^3 multiplied by 16a^8 and yielding the final answer.
Step-by-step explanation:
The question asks us to expand the expression (3a)^3(4a^4)^2 without using exponents and to evaluate it. We start by applying the rule for cubing of exponentials to the first term and the rule of powers to the second term. When we cube a term like (3a)^3, we multiply the base, 3, by itself three times and the exponent, 1 (implied for a), by 3. Similarly, when we square (4a^4)^2, we multiply the base, 4, by itself once and the exponent, 4, by 2.
Expanding the terms, we get
(3a)^3 = 3^3 * a^(1*3) = 27a^3
(4a^4)^2 = 4^2 * a^(4*2) = 16a^8
Now we multiply the expanded forms:
27a^3 * 16a^8
To multiply terms with the same base (in this case, a), we simply add their exponents. So we get:
27 * 16 * a^(3+8) = 432 * a^11
Therefore, the expression (3a)^3(4a^4)^2 expanded and evaluated is 432a^11.