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To begin tackling the exercise, we first need to understand the power rule for exponents. This rule is vital when dealing with a power raised to another power. The rule is expressed as \((a^m)^n = a^{m \times n}\). Whenever you see a base raised to an exponent that is itself raised to another exponent, multiply the exponents together.

In our exercise, we have the term \((a^3)^2\). According to the power rule, this simplifies to \((a^{3 \times 2}) = a^6\).

Utilizing the power rule allows us to simplify complex exponent expressions step by step, making the process much more manageable.

The next step for simplifying exponents involves the quotient rule. This rule is used when dealing with the division of two expressions with the same base and different exponents. It is expressed as \(\frac{a^m}{a^n} = a^{m-n}\).

Once we've applied the power rule, we have the expression \(\frac{a^6}{a^4}\). According to the quotient rule, we subtract the exponent of the denominator from the exponent of the numerator: \(\frac{a^6}{a^4} = a^{6-4} = a^2\).

The quotient rule is essential for simplifying fractions that involve exponents, allowing us to reduce them to their simplest form quickly.

Simplifying fractions, especially those involving exponents, makes solving equations significantly easier. After applying both the power rule and the quotient rule, we simplify our original complex expression to a much more manageable form.

Initially, the expression was \(\frac{\big(a^3\big)^2}{a^4}\). After applying the power rule, it became \(\frac{a^6}{a^4}\). Using the quotient rule further simplified it to \((a^2)\), which is the simplest form.

Practicing with different equations helps build a solid understanding of how to simplify fractions involving exponents. Remember to always break down the problem step by step, applying the necessary rules, and you will find the process much easier.