To simplify the expression \[ \frac{3(2^{n+1}) - 4(2^{n-1})}{2^{n+1} - 2^n} \] let's break it down step by step. ### Step 1: Simplify the Numerator The numerator is \[ 3(2^{n+1}) - 4(2^{n-1}). \] We can rewrite \(2^{n+1}\) and \(2^{n-1}\) in terms of \(2^n\): - \(2^{n+1} = 2 \cdot 2^n\) - \(2^{n-1} = \frac{2^n}{2}\) Substituting these into the numerator gives: \[ 3(2 \cdot 2^n) - 4\left(\frac{2^n}{2}\right) = 6 \cdot 2^n - 2 \cdot 2^n = (6 - 2) \cdot 2^n = 4 \cdot 2^n. \] ### Step 2: Simplify the Denominator The denominator is \[ 2^{n+1} - 2^n. \] Using the same substitution as before: \[ 2^{n+1} - 2^n = 2 \cdot 2^n - 2^n = (2 - 1) \cdot 2^n = 1 \cdot 2^n = 2^n. \] ### Step 3: Combine the Simplified Parts Now we can substitute the simplified numerator and denominator back into the fraction: \[ \frac{4 \cdot 2^n}{2^n}. \] ### Step 4: Cancel Common Terms Since \(2^n\) is in both the numerator and the denominator, we can cancel it out: \[ \frac{4 \cdot 2^n}{2^n} = 4. \] ### Conclusion Thus, the simplified expression is \[ 4. \] ### Answer The correct option is **C. 4**. ### Explanation of Other Options - **A. 2n + 1**: This option suggests a linear expression in terms of \(n\). However, our simplification led us to a constant value of 4, not a function of \(n\). - **B. 2n - 1**: Similar to option A, this is also a linear expression. The simplification does not yield a linear function of \(n\), so this option is incorrect. - **D. 1/4**: This option suggests a fraction that is much smaller than our result. Since we found the simplified expression to be 4, this option is also incorrect. ### Revision Summary - Simplify the numerator and denominator separately before combining. - Use properties of exponents to rewrite terms for easier manipulation. - Cancel common factors in the numerator and denominator to simplify the expression. - The final simplified result of the given expression is **4**.