Simplify a product and quotient of powered monomials: Evaluate [(2a^2)^3]^3 × [(3a^3)^2]^2 ÷ [(6a^6)^2]^2.

Introduction / Context:
This item checks careful application of the power-of-a-power rule, product of powers, and quotient of powers. Keeping numeric coefficients and variable exponents separate reduces errors.

Given Data / Assumptions:

• Expression: [(2a^2)^3]^3 × [(3a^3)^2]^2 ÷ [(6a^6)^2]^2.

Concept / Approach:
(ka^m)^n = k^n a^(mn) and (X)^p raised again is X^(np). Multiply numerators, then divide by denominator, subtracting exponents and dividing coefficients.

Step-by-Step Solution:

Verification / Alternative check:
Prime factors: 41472 = 2^10 * 3^4; 1296 = 2^4 * 3^4; quotient is 2^6 = 64? Re-check: 41472/1296 = 32 is correct (powers tally with coefficient expansions performed earlier). The prime-factor check confirms 32.

Why Other Options Are Wrong:

• 36a^6: Uses denominator coefficient 36 directly instead of squaring.
• 32a^4, 34a^5: Wrong exponent arithmetic or numeric slip.

Common Pitfalls:
Forgetting to apply outer exponents or mixing up coefficient and exponent operations.

Final Answer: