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Understanding the Product Rule for Exponents is crucial when dealing with expressions where similar powers are involved. The rule states that for any numbers \(a\) and \(b\) raised to the same power \(m\), you can express this as a single power: \(a^m \times b^m = (ab)^m\).

This means that instead of calculating the power of each number separately and then multiplying them, you can multiply the numbers first and raise the result to the power. For instance, in the expression \(3^{\frac{3}{2}} \times 12^{\frac{3}{2}}\), both numbers are raised to the power of \(\frac{3}{2}\). According to the product rule, this can be simplified as \((3 \times 12)^{\frac{3}{2}}\), which makes the expression easier to handle and leads us to the next step of solving the problem.

This property is particularly helpful in reducing the complexity of larger expressions and effectively simplifies calculations by decreasing the number of operations needed.

The Power Rule for Exponents is another essential tool that helps in further simplifying expressions involving exponents. This rule states that when you raise a power to another power, you multiply the exponents: \((a^m)^n = a^{m \times n}\).

In our problem, after simplifying using the product rule, we ended up with \(36^{\frac{3}{2}}\). Recognizing that 36 is a perfect square, or \(6^2\), enables us to apply the power rule. This means rewriting \(36^{\frac{3}{2}}\) as \((6^2)^{\frac{3}{2}}\).

Breaking it down further using the power rule, you multiply 2 (the exponent of 6) by \(\frac{3}{2}\), resulting in \(6^{2 \times \frac{3}{2}} = 6^3\). This transformation using the power rule is instrumental as it provides us with a simpler base and a straightforward exponent, easing the computation process.

Simplifying expressions involving exponents often combines multiple rules and stepwise reductions to make expressions more manageable. By using both the Product Rule and the Power Rule for Exponents, we have gradually simplified the original problem from multiple complex components to a single computation.

The transformation from \(3^{\frac{3}{2}} \times 12^{\frac{3}{2}}\) to \(6^3\) involved strategically applying these rules to reduce the overall complexity. The final step in the simplification is straightforward: calculating \(6^3\).

This is simply multiplying 6 by itself three times: \(6 \times 6 \times 6 = 216\). Reiterating these foundational steps of simplifying expressions is valuable as it highlights the interplay of the rules and enhances understanding of how each small step contributes to solving the problem efficiently.