Solve for the exponent: If 289 = 17^(x/5), find the value of x.

Introduction / Context:Recognize perfect powers: 289 is 17 squared. Equating exponents on the same base determines x straightforwardly.

Given Data / Assumptions:

• 289 = 17^(x/5)
• 17^2 = 289

Concept / Approach:Rewrite 289 as 17^2. Then match exponents: x/5 = 2, solve for x by multiplying both sides by 5.

Step-by-Step Solution:17^(x/5) = 17^2 ⇒ x/5 = 2x = 2 * 5 = 10

Verification / Alternative check:Substitute back: 17^(10/5) = 17^2 = 289, correct.

Why Other Options Are Wrong:They result from mis-scaling the fraction or confusing 289 with another square.

Common Pitfalls:Forgetting to multiply by 5; ensure exact base matching before equating exponents.

Final Answer:10