Simplify the expression: \(\left(\frac{3m+1}{3m-1}-\frac{3m-1}{3m+1}\right):\frac{4m}{9m+3}\)

This is a math problem involving simplification of algebraic fractions. We need to perform the subtraction within the parentheses first, then the division.

1. Combine fractions inside the parentheses: Find a common denominator, which is \( (3m-1)(3m+1) \).
• \[ \frac{3m+1}{3m-1} - \frac{3m-1}{3m+1} = \frac{(3m+1)^2 - (3m-1)^2}{(3m-1)(3m+1)} \]
• Expand the squares:
• \[ (3m+1)^2 = (3m)^2 + 2(3m)(1) + 1^2 = 9m^2 + 6m + 1 \]
• \[ (3m-1)^2 = (3m)^2 - 2(3m)(1) + 1^2 = 9m^2 - 6m + 1 \]
• Substitute back into the expression:
• \[ \frac{(9m^2 + 6m + 1) - (9m^2 - 6m + 1)}{(3m-1)(3m+1)} = \frac{9m^2 + 6m + 1 - 9m^2 + 6m - 1}{9m^2 - 1} = \frac{12m}{9m^2 - 1} \]
2. Perform the division: Dividing by a fraction is the same as multiplying by its reciprocal.
• The expression becomes:
• \[ \frac{12m}{9m^2 - 1} \div \frac{4m}{9m+3} = \frac{12m}{9m^2 - 1} \times \frac{9m+3}{4m} \]
• Factor the terms:
• \[ 9m^2 - 1 = (3m-1)(3m+1) \]
• \[ 9m+3 = 3(3m+1) \]
• Substitute the factored terms back into the expression:
• \[ \frac{12m}{(3m-1)(3m+1)} \times \frac{3(3m+1)}{4m} \]
• Cancel out common factors: \( 12m \) in the numerator and \( 4m \) in the denominator result in 3 in the numerator. \( (3m+1) \) in the numerator and denominator cancel out.
• The simplified expression is:
• \[ \frac{3 imes 3}{3m-1} = \frac{9}{3m-1} \]

Ответ:

\[ \frac{9}{3m-1} \]