a) \( \frac{8^{n} q^{n}}{2^{3 n}-3^{n}} \)

To simplify the expression \( \frac{8^{n} q^{n}}{2^{3 n}-3^{n}} \), we start by rewriting \( 8^{n} \) in terms of base 2: \[ 8^{n} = (2^{3})^{n} = 2^{3n} \] Now we can substitute this back into the expression: \[ \frac{8^{n} q^{n}}{2^{3 n}-3^{n}} = \frac{2^{3n} q^{n}}{2^{3n} - 3^{n}} \] Next, we can factor \( 2^{3n} \) out from the denominator: \[ 2^{3n} - 3^{n} = 2^{3n}(1 - \frac{3^{n}}{2^{3n}}) \] Now the expression looks like this: \[ \frac{2^{3n} q^{n}}{2^{3n}(1 - \frac{3^{n}}{2^{3n}})} = \frac{q^{n}}{1 - \frac{3^{n}}{2^{3n}}} \] Therefore, the simplified expression is: \[ \frac{q^{n}}{1 - \left(\frac{3}{8}\right)^{n}} \]