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When simplifying algebraic expressions, one of the first steps is often exponent distribution. This process involves applying an exponent to all bases within a set of parentheses. Consider the expression \( (r^{3} s^{2})^{2} \). Here, the exponent \(2\) outside the parentheses must be distributed to both \(r^{3}\) and \(s^{2}\), resulting in \(r^{3*2}\) and \(s^{2*2}\), which simplifies to \(r^{6}\) and \(s^{4}\) respectively.

It is crucial to understand that exponent distribution requires multiplying the exponents, not adding them. This is based on the rule of powers which states that \( (a^{m})^{n} = a^{m*n} \). Applying this rule carefully is essential for correctly simplifying expressions that involve exponents.

After distributing the exponents, attention shifts to coefficient simplification. A coefficient is a numerical factor that multiplies a variable or a product of variables. In our expression, the coefficients are the numbers 35 and 49. These can often be simplified by dividing them by their greatest common divisor. In the expression \( \frac{35}{49} \), both numbers are divisible by 7, so when you divide them, you obtain the simplified fraction \( \frac{5}{7} \).

Coefficient simplification is like reducing fractions; it's about making the numbers as small and manageable as possible without changing the value of the expression. Always look for common factors between coefficients to simplify the expression to its most basic form.

Variable exponent subtraction is applied when dividing like variables with exponents. In the expression \( \frac{5 r^6 s^4}{7 r^2 s^3} \), notice that both the numerator and denominator contain the variables \(r\) and \(s\) raised to a power. To simplify, subtract the exponent of the variable in the denominator from the exponent of the same variable in the numerator. For \(r\), subtract 2 from 6 resulting in \(r^{6-2} = r^4\). For \(s\), subtract 3 from 4 resulting in \(s^{4-3} = s\).

This operation is grounded on the quotient of powers rule, which says \(a^{m} / a^{n} = a^{m-n}\), provided that \(a eq 0\). Simplifying expressions in this manner leads to easier evaluation and understanding of algebraic expressions.