Final answer:
To solve the equation, exponent rules are applied to simplify the expression, which leads to the conclusion that x equals 2. Therefore, the correct answer is (c) x = 2.
Step-by-step explanation:
To solve for x in the equation (2^3 \times 3^5)^2 \div (2^4 \times 3^6) \div 3^2 = 6^x, we first simplify the given equation. According to the rules of exponents, when we raise an exponentiated quantity to a power, we multiply the exponents.
For example, ((a^b)^c = a^(b \cdot c)), so (2^3 \times 3^5)^2 becomes 2^(3\cdot2) \times 3^(5\cdot2) which is 2^6 \times 3^10. When we divide exponents with the same base, we subtract the exponents (x^a \div x^b = x^(a-b)).
Therefore, we have:
2^6 \times 3^10 \div 2^4 \times 3^6 \div 3^2
This simplifies to 2^(6-4) \times 3^(10-6-2) which is 2^2 \times 3^2. Note that 6^x equates to (2 \times 3)^x, which is 2^x \times 3^x.
Therefore, we have 2^2 \times 3^2 = 2^x \times 3^x. Matching the exponents on both sides of the equal sign gives us x = 2.
The correct answer is (c) x = 2.