To simplify the expression \(\frac{6 m^2 n^2}{3 n}\), we can follow these detailed steps:
1. Separate the Coefficients from the Variables:
- Identify the coefficients and the variables in the numerator and the denominator.
- In the numerator, we have the coefficient 6, and the variables \(m^2\) and \(n^2\).
- In the denominator, we have the coefficient 3 and the variable \(n\).
2. Simplify the Coefficients:
- Divide the coefficient in the numerator by the coefficient in the denominator: \(\frac{6}{3} = 2\).
3. Simplify the Exponents of \(m\):
- In the numerator, we have \(m^2\), and there is no \(m\) term in the denominator, so \(m^2\) remains as it is.
4. Simplify the Exponents of \(n\):
- In the numerator, we have \(n^2\).
- In the denominator, we have \(n\).
- Apply the rule of exponents where \(\frac{n^a}{n^b} = n^{a-b}\).
- Thus, \(\frac{n^2}{n} = n^{2-1} = n^1 = n\).
5. Combine the Simplified Terms:
- After simplification, the coefficient is 2.
- The \(m\) term remains as \(m^2\).
- The \(n\) term is simplified to \(n\).
Therefore, the simplified result is:
[tex]\[
\frac{6 m^2 n^2}{3 n} = 2 m^2 n
\][/tex]
