The function r is defined by \(\mathrm{r(k) = 180[(0.6)^k + (0.2)^k]}\). What is the value of \(\mathrm{r(0)}\)?

1. TRANSLATE the problem requirement

• Given: \(\mathrm{r(k) = 180[(0.6)^k + (0.2)^k]}\)
• Find: \(\mathrm{r(0)}\)
• This means: substitute \(\mathrm{k = 0}\) into the function

2. TRANSLATE the substitution

• Replace every k with 0:

\(\mathrm{r(0) = 180[(0.6)^0 + (0.2)^0]}\)

3. INFER which mathematical rule applies

• Both terms have exponent 0
• The zero exponent rule applies: any non-zero number to the power of 0 equals 1
• Therefore: \(\mathrm{(0.6)^0 = 1}\) and \(\mathrm{(0.2)^0 = 1}\)

4. SIMPLIFY the expression

• \(\mathrm{r(0) = 180[1 + 1]}\)
• \(\mathrm{r(0) = 180[2]}\)
• \(\mathrm{r(0) = 360}\)

Answer: D (360)

Why Students Usually Falter on This Problem

Most Common Error Path:

Missing conceptual knowledge: Zero exponent rule

Students may not remember or recognize that any non-zero number raised to the power of 0 equals 1. Instead, they might think \(\mathrm{(0.6)^0 = 0.6}\) or \(\mathrm{(0.2)^0 = 0.2}\), leading to \(\mathrm{r(0) = 180[0.6 + 0.2] = 180[0.8] = 144}\).

This may lead them to select Choice B (144).

Second Most Common Error:

Weak INFER skill: Confusing zero exponent with multiplication by zero

Some students might incorrectly think that raising to the power of 0 means the result is 0, giving them \(\mathrm{(0.6)^0 = 0}\) and \(\mathrm{(0.2)^0 = 0}\). This leads to \(\mathrm{r(0) = 180[0 + 0] = 180[0] = 0}\).

This may lead them to select Choice A (0).

The Bottom Line:

This problem tests whether students have memorized and can apply the zero exponent rule. Without this fundamental concept, even students who correctly translate the problem will arrive at wrong answers.