\( \frac { ( \frac { 1 ^ { \prime \prime } n ^ { \prime \prime } } { 5 } - \frac { 2 } { 3 ^ { \prime \prime } n ^ { \prime \prime } } ) ^ { 3 } } { \frac { 3 } { 2 ^ { \prime \prime } n ^ { \prime \prime } } x \frac { 1 n ^ { \prime \prime } } { 9 } } \)

To simplify the expression \[ \frac { ( \frac { 1 ^ { \prime \prime } n ^ { \prime \prime } } { 5 } - \frac { 2 } { 3 ^ { \prime \prime } n ^ { \prime \prime } } ) ^ { 3 } } { \frac { 3 } { 2 ^ { \prime \prime } n ^ { \prime \prime } } x \frac { 1 n ^ { \prime \prime } } { 9 } } \] we first need to clarify the notation. The notation \( n^{\prime\prime} \) is not standard, but it seems to indicate some variable \( n \) with a double prime. For the sake of simplification, we will treat \( n^{\prime\prime} \) as just \( n \). Now, let's rewrite the expression: \[ \frac { \left( \frac { 1 n } { 5 } - \frac { 2 } { 3 n } \right) ^ { 3 } } { \frac { 3 } { 2 n } x \cdot \frac { 1 n } { 9 } } \] Next, we simplify the numerator: 1. Find a common denominator for the terms in the parentheses: - The common denominator of \( 5 \) and \( 3n \) is \( 15n \). - Rewrite the first term: \( \frac{1n}{5} = \frac{3n}{15n} \). - Rewrite the second term: \( \frac{2}{3n} = \frac{10}{15n} \). Thus, we have: \[ \frac{3n}{15n} - \frac{10}{15n} = \frac{3n - 10}{15n} \] Now, we raise this to the power of 3: \[ \left( \frac{3n - 10}{15n} \right)^3 = \frac{(3n - 10)^3}{(15n)^3} = \frac{(3n - 10)^3}{3375n^3} \] Now, let's simplify the denominator: \[ \frac{3}{2n} x \cdot \frac{1n}{9} = \frac{3 \cdot 1n \cdot x}{2n \cdot 9} = \frac{3x}{18} = \frac{x}{6} \] Now we can rewrite the entire expression: \[ \frac{\frac{(3n - 10)^3}{3375n^3}}{\frac{x}{6}} = \frac{(3n - 10)^3}{3375n^3} \cdot \frac{6}{x} = \frac{6(3n - 10)^3}{3375n^3 x} \] Thus, the simplified expression is: \[ \frac{6(3n - 10)^3}{3375n^3 x} \]