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Understanding the cube exponent rule is crucial when dealing with algebraic expressions that involve raising terms to the power of three. The 'cube' of a number or a variable is simply that quantity multiplied by itself three times. For example, the cube of 2 is \(2 \times 2 \times 2 = 8\).

When using the cube exponent rule to an algebraic expression, such as \( (b^{2})^{3} \), you are essentially calculating \( b^{2} \cdot b^{2} \cdot b^{2} \), which simplifies to \( b^{6} \) since you are multiplying the base \( b \) by itself a total of six times (two times for each cube).

Thus, in the context of our exercise, the cube rule dictates that each element within the parentheses is raised to the third power. Once applied, it transforms \( \left(\frac{b^{2}}{3a}\right)^{3} \) into \( (b^{2})^{3} \) and \( (3a)^{3} \) respectively for the numerator and the denominator.

The rule of exponents provides a framework for simplifying expressions that involve powers. When you have an exponent raised to another exponent, like \( (b^{2})^{3} \), you multiply the exponents. This rule is summarized as \( (b^{m})^{n} = b^{m \cdot n} \).

So, when simplifying the numerator of our given problem, \( (b^{2})^{3} \), we follow this rule and multiply 2 (the power of \( b \)) by 3 (the external exponent), resulting in \( b^{2 \cdot 3} = b^{6} \). This process is crucial for correctly manipulating exponents and forms the basis for simplifying our original algebraic expression.

Simplifying fractions involves reducing them to their simplest form. Algebraic fractions are simplified by applying rules for exponents and performing arithmetic operations on the coefficients and the variable terms separately.

For the solution of our expression \( \left(\frac{b^{2}}{3a}\right)^{3} \), after applying the cube rule and the rule of exponents, we are left with \( \frac{b^{6}}{27a^{3}} \). Here, we've already simplified the fraction by cubing both the numerator and denominator independently. There are no common factors between the numerator and the denominator, so the fraction is in its simplest form.

Sometimes, additional steps like factoring or finding common denominators are needed to further reduce fractions. However, in our case, since \( b \) and \( a \) are different variables, and 27 does not factor any further with respect to the numerator, no additional simplification is necessary.