The value of $\frac{2^{3x+4} + 8^{x+1}}{8^{x+1} - 2^{3x+2}}$ is

### Concept & Formula This question requires converting different bases to a common prime base (2 in this case) and subsequently factoring out terms to simplify the fraction. The governing exponent rule is: $$ (a^m)^n = a^{m \times n} $$ ### Step-by-Step Solution * **Step 1:** Convert the base 8 terms to base 2. We know $8 = 2^3$. $8^{x+1} = (2^3)^{x+1} = 2^{3(x+1)} = 2^{3x+3}$ * **Step 2:** Substitute this converted term back into the original fraction. Numerator: $2^{3x+4} + 2^{3x+3}$ Denominator: $2^{3x+3} - 2^{3x+2}$ * **Step 3:** Factor out the smallest power of 2 from both the numerator and the denominator. Numerator: Factor out $2^{3x+3}$ $2^{3x+3} (2^1 + 1) = 2^{3x+3} (3)$ Denominator: Factor out $2^{3x+2}$ $2^{3x+2} (2^1 - 1) = 2^{3x+2} (1)$ * **Step 4:** Reconstruct the fraction with the factored expressions. $\frac{2^{3x+3} \times 3}{2^{3x+2}}$ * **Step 5:** Simplify the exponent term using the division law. $2^{(3x+3) - (3x+2)} \times 3$ $2^1 \times 3 = 6$ ### Exam Strategy & Shortcut **Variable Substitution:** Since the options are all constants, the expression's value is independent of $x$. Choose the simplest value for $x$, which is $x = 0$. Substitute $x = 0$: Numerator: $2^4 + 8^1 = 16 + 8 = 24$ Denominator: $8^1 - 2^2 = 8 - 4 = 4$ Result: $\frac{24}{4} = 6$. This method resolves the problem in roughly 10 seconds without needing to apply index laws. ### Common Pitfall A common mistake when applying the exponent rule $(a^m)^n$ is failing to distribute the multiplier across the entire binomial. Students might write $(2^3)^{x+1}$ as $2^{3x+1}$ instead of the correct $2^{3x+3}$, which cascades into a completely wrong result. ### Final Answer **Therefore, the correct answer is 6.**