6. \( \frac{x^{2 n+1} \cdot x^{2}}{x^{n+3}} \)

To simplify the expression \( \frac{x^{2n+1} \cdot x^{2}}{x^{n+3}} \), we can start by combining the terms in the numerator. The numerator \( x^{2n+1} \cdot x^{2} \) can be rewritten using the property of exponents that states \( x^a \cdot x^b = x^{a+b} \). Thus, we get: \[ x^{(2n+1) + 2} = x^{2n + 3} \] Now, substituting this into the original expression: \[ \frac{x^{2n + 3}}{x^{n + 3}} \] Next, applying the quotient rule for exponents, which says \( \frac{x^a}{x^b} = x^{a-b} \), gives us: \[ x^{(2n + 3) - (n + 3)} = x^{2n + 3 - n - 3} = x^{n} \] Thus, the simplified form of the expression is: \[ \boxed{x^{n}} \]