Simplify the following expression: \(\frac{3n^6 + 2n^4}{15n^8 + 10n^6}\)

Simplification of a Rational Expression

Steps:

1. Step 1: Factor the numerator.
   The numerator is \(3n^6 + 2n^4\). The GCF of the terms \(3n^6\) and \(2n^4\) is \(n^4\).
   Factoring out \(n^4\) gives: \(n^4(3n^2 + 2)\).
2. Step 2: Factor the denominator.
   The denominator is \(15n^8 + 10n^6\). The GCF of the terms \(15n^8\) and \(10n^6\) is \(5n^6\).
   Factoring out \(5n^6\) gives: \(5n^6(3n^2 + 2)\).
3. Step 3: Rewrite the expression with factored terms.
   The expression becomes \(\frac{n^4(3n^2 + 2)}{5n^6(3n^2 + 2)}\).
4. Step 4: Cancel out common factors.
   The term \((3n^2 + 2)\) is common to both the numerator and the denominator, so it can be canceled out.
   The expression simplifies to \(\frac{n^4}{5n^6}\).
5. Step 5: Simplify the remaining terms.
   Using the rule of exponents \(\frac{a^m}{a^n} = a^{m-n}\), we can simplify \(\frac{n^4}{n^6}\) to \(n^{4-6} = n^{-2} = \frac{1}{n^2}\).
   Therefore, the simplified expression is \(\frac{1}{5n^2}\).

Final Answer: \(\frac{1}{5n^2}\)