Solve the exponential sum: If 2^(x−1) + 2^(x+1) = 2560, find the value of x.

Introduction / Context:This evaluates comfort with factoring common powers in exponential equations. Recognizing a shared factor in 2^(x−1) and 2^(x+1) simplifies the equation greatly.

Given Data / Assumptions:

• 2^(x−1) + 2^(x+1) = 2560
• 2 is the common base.

Concept / Approach:Factor out the smaller power 2^(x−1). Use 2^(x+1) = 2^(x−1)*2^2 to combine terms. Then solve a simple power-of-two equation by matching exponents.

Step-by-Step Solution:2^(x−1) + 2^(x+1) = 2^(x−1)(1 + 2^2) = 2^(x−1)*5So 2^(x−1)*5 = 2560 ⇒ 2^(x−1) = 2560 / 5 = 512512 = 2^9 ⇒ x − 1 = 9 ⇒ x = 10

Verification / Alternative check:Check: 2^9 + 2^11 = 512 + 2048 = 2560, correct.

Why Other Options Are Wrong:Values like 8, 9, 11, 12 do not satisfy the exact power equality when substituted.

Common Pitfalls:Factoring out 2^x instead of 2^(x−1) is fine but gives fractions; choosing the smallest exponent keeps arithmetic clean.

Final Answer:10