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

Certainly! Let's simplify the expression you've provided: \[ \frac{8^{n} q^{n}}{2^{3n} - 3^{n}} \] ### Step 1: Simplify the Numerator First, recognize that \(8^n\) can be rewritten in terms of powers of 2: \[ 8^n = (2^3)^n = 2^{3n} \] So, the numerator becomes: \[ 8^{n} q^{n} = 2^{3n} q^{n} \] ### Step 2: Analyze the Denominator The denominator is already expressed as: \[ 2^{3n} - 3^{n} \] ### Step 3: Rewrite the Entire Expression Now, substitute the simplified numerator back into the original expression: \[ \frac{2^{3n} q^{n}}{2^{3n} - 3^{n}} \] ### Step 4: Factor Out \(2^{3n}\) from the Denominator To further simplify, factor \(2^{3n}\) out of the denominator: \[ 2^{3n} - 3^{n} = 2^{3n} \left(1 - \frac{3^{n}}{2^{3n}}\right) = 2^{3n} \left(1 - \left(\frac{3}{8}\right)^n\right) \] ### Step 5: Simplify the Fraction Now, substitute this back into the expression: \[ \frac{2^{3n} q^{n}}{2^{3n} \left(1 - \left(\frac{3}{8}\right)^n\right)} = \frac{q^{n}}{1 - \left(\frac{3}{8}\right)^n} \] ### Final Simplified Form \[ \frac{q^{n}}{1 - \left(\frac{3}{8}\right)^n} \] ### Interpretation This simplified form expresses the original fraction in terms of \(q^n\) and a geometric factor \(\left(\frac{3}{8}\right)^n\). This can be particularly useful for analyzing the behavior of the expression as \(n\) grows large or for further algebraic manipulations, depending on the context of the problem. If you have more context or further parts to the problem, feel free to share them, and I can assist you further!