Question:The function f is defined by the given equation. The equation can be rewritten as \(\mathrm{f(x) = (1 + p/100)}^{\mathrm{x}}\),...

1. TRANSLATE the problem information

• Given: \(\mathrm{f(x) = (1.05)^{4x}}\)
• Need to find: Value of p where \(\mathrm{f(x) = (1 + p/100)^x}\)
• This means we need these two expressions to be equivalent for all x-values

2. INFER the key relationship

• For two exponential expressions to be equal for ALL x-values, their bases must be equal
• Since \(\mathrm{(1.05)^{4x} = ((1.05)^4)^x}\), we need:
  \(\mathrm{(1.05)^4 = 1 + p/100}\)

3. SIMPLIFY to find the base value

• Calculate \(\mathrm{(1.05)^4}\):
  • First: \(\mathrm{(1.05)^2 = 1.1025}\)
  • Then: \(\mathrm{(1.05)^4 = (1.1025)^2 = 1.21550625}\) (use calculator)

4. SIMPLIFY to solve for p

• Set up the equation: \(\mathrm{1 + p/100 = 1.21550625}\)
• Subtract 1: \(\mathrm{p/100 = 0.21550625}\)
• Multiply by 100: \(\mathrm{p = 21.550625}\)

5. APPLY CONSTRAINTS to select final answer

• Looking at answer choices: (A) 10, (B) 16, (C) 21, (D) 46
• Our calculated value 21.55 is closest to 21

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may not recognize that the bases must be equal for exponential equations to be equivalent for all x-values. Instead, they might try to manipulate the exponents directly or attempt to substitute specific x-values, leading to confusion about how to proceed systematically. This leads to abandoning the systematic approach and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify that \(\mathrm{(1.05)^4 = 1 + p/100}\), but make calculation errors when computing \(\mathrm{(1.05)^4}\). Common mistakes include:

• Computing \(\mathrm{(1.05)^2}\) incorrectly as 1.10 instead of 1.1025
• Making arithmetic errors when squaring 1.1025
• Incorrectly solving \(\mathrm{p/100 = 0.21550625}\) for p

These calculation errors could lead them to values closer to other answer choices, causing them to select Choice A (10) or Choice B (16).

The Bottom Line:

This problem requires recognizing that exponential function equivalence depends on base equality, combined with careful multi-step calculations. The key insight is understanding when two exponential expressions represent the same function.