The function f is defined by \(\mathrm{f(x) = 270(0.1)^x}\). What is the value of \(\mathrm{f(0)}\)?

1. TRANSLATE the question into mathematical operations

• Given information:
  • Function: \(\mathrm{f(x) = 270(0.1)^x}\)
  • Need to find: \(\mathrm{f(0)}\)

• What this means: Substitute \(\mathrm{x = 0}\) into the function and calculate the result

2. INFER the approach needed

• To find \(\mathrm{f(0)}\), substitute 0 for x in the function
• This gives us: \(\mathrm{f(0) = 270(0.1)^0}\)
• The key insight: We need to evaluate \(\mathrm{(0.1)^0}\) using the zero exponent rule

3. SIMPLIFY using the zero exponent rule

• Apply the rule: Any non-zero number raised to the power 0 equals 1
• So \(\mathrm{(0.1)^0 = 1}\)
• Therefore: \(\mathrm{f(0) = 270(1) = 270}\)

Answer: D. 270

Why Students Usually Falter on This Problem

Most Common Error Path:

Missing conceptual knowledge: Not knowing or forgetting the zero exponent rule

Students might think that \(\mathrm{(0.1)^0 = 0.1}\) or even that \(\mathrm{(0.1)^0 = 0}\), leading to incorrect calculations like \(\mathrm{f(0) = 270(0.1) = 27}\) or \(\mathrm{f(0) = 270(0) = 0}\).

This may lead them to select Choice C (27) or Choice A (0).

The Bottom Line:

This problem tests whether students remember that any non-zero number raised to the power 0 always equals 1, regardless of what the base number is. The zero exponent rule is a fundamental property that students often forget or confuse with other exponent rules.