The function p is defined by \(\mathrm{p(n) = 7n^3}\). What is the value of n when \(\mathrm{p(n)}\) is equal to...

1. TRANSLATE the problem information

• Given information:
  • Function definition: \(\mathrm{p(n) = 7n^3}\)
  • Condition: \(\mathrm{p(n) = 56}\)
  • Need to find: value of n

• What this tells us: We need to substitute 56 for p(n) in the function equation

2. SIMPLIFY to set up the equation

• Since \(\mathrm{p(n) = 7n^3}\) and \(\mathrm{p(n) = 56}\), we can write:
  \(\mathrm{56 = 7n^3}\)

3. SIMPLIFY to isolate n³

• Divide both sides by 7:
  \(\mathrm{56 ÷ 7 = 7n^3 ÷ 7}\)
  \(\mathrm{8 = n^3}\)

4. SIMPLIFY to find n

• Take the cube root of both sides:
  \(\mathrm{\sqrt[3]{8} = \sqrt[3]{n^3}}\)
  \(\mathrm{2 = n}\)

Answer: A. 2

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students confuse cube root with square root when solving \(\mathrm{n^3 = 8}\)

Instead of recognizing that \(\mathrm{\sqrt[3]{8} = 2}\), they might think \(\mathrm{n^2 = 8}\), so \(\mathrm{n = \sqrt{8} ≈ 2.83}\). This could lead them toward Choice B (8/3), which is approximately 2.67.

Second Most Common Error:

Incomplete SIMPLIFY process: Students correctly get to \(\mathrm{n^3 = 8}\) but then think the answer is 8

They stop at \(\mathrm{n^3 = 8}\) and assume \(\mathrm{n = 8}\), not realizing they need to take the cube root. This leads them to select Choice D (8).

The Bottom Line:

This problem tests whether students can work backwards from a function output to find the input, requiring careful execution of inverse operations. The key challenge is remembering that to "undo" a cube operation, you need a cube root, not a square root.