\(\mathrm{f(x) = (1.84)^{x/4}}\) The function f is defined by the given equation. The equation \(\mathrm{f(x) = (1 + p/100)^x}\), where...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{f(x) = (1.84)^{(x/4)}}\)
  • \(\mathrm{f(x) = (1 + p/100)^x}\) (equivalent form)
  • Need to find p
• What this tells us: We need to rewrite the first expression to match the form of the second expression.

2. INFER the approach

• Since both expressions equal f(x), they must be equivalent
• The second expression has the form \(\mathrm{(base)^x}\), so I need to rewrite the first expression in this same form
• This means converting \(\mathrm{(1.84)^{(x/4)}}\) into something like \(\mathrm{(base)^x}\)

3. SIMPLIFY the exponential expression

• Rewrite \(\mathrm{(1.84)^{(x/4)}}\) using exponent properties:
  \(\mathrm{(1.84)^{(x/4)} = (1.84)^{(1/4·x)} = ((1.84)^{(1/4)})^x}\)
• Calculate the base: \(\mathrm{(1.84)^{(1/4)} ≈ 1.16467}\) (use calculator)
• So \(\mathrm{f(x) = (1.16467)^x}\)

4. INFER the relationship between bases

• Since \(\mathrm{f(x) = (1.16467)^x}\) and \(\mathrm{f(x) = (1 + p/100)^x}\)
• The bases must be equal: \(\mathrm{1 + p/100 = 1.16467}\)

5. SIMPLIFY to solve for p

• \(\mathrm{1 + p/100 = 1.16467}\)
• \(\mathrm{p/100 = 0.16467}\)
• \(\mathrm{p = 16.467}\)
• Checking the choices: A. 16, B. 21, C. 46, D. 96
• 16 is closest to 16.467

Answer: A. 16

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students struggle with rewriting \(\mathrm{(1.84)^{(x/4)}}\) in the correct form. They might attempt to work directly with the fractional exponent without recognizing they need to use exponent properties to create a base raised to the power x.

Common wrong approach: Trying to set \(\mathrm{1.84 = 1 + p/100}\) and solving \(\mathrm{p = 84}\), which doesn't match any answer choice, leading to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the need to find \(\mathrm{(1.84)^{(1/4)}}\) but make calculation errors, either by estimating incorrectly or making arithmetic mistakes when converting to percentage form.

For example, if they calculate \(\mathrm{(1.84)^{(1/4)} ≈ 1.21}\) (incorrect), they would get \(\mathrm{p ≈ 21}\), leading them to select Choice B (21).

The Bottom Line:

This problem tests whether students can manipulate exponential expressions using properties of exponents to create equivalent forms. The key insight is recognizing that \(\mathrm{(a^b)^c = a^{(bc)}}\), which allows the conversion from fractional to whole number exponents.