The function g is defined by \(\mathrm{g(x) = 5\sqrt[3]{x}}\). For what value of x does \(\mathrm{g(x) = 15}\)?392745

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{g(x) = 5\sqrt[3]{x}}\)
  • We need: \(\mathrm{g(x) = 15}\)

• This tells us we need to solve: \(\mathrm{5\sqrt[3]{x} = 15}\)

2. SIMPLIFY to isolate the cube root

• Divide both sides by 5 to get the cube root by itself:
  \(\mathrm{\sqrt[3]{x} = 15 \div 5 = 3}\)

• Now we have: \(\mathrm{\sqrt[3]{x} = 3}\)

3. SIMPLIFY to solve for x

• To eliminate the cube root, cube both sides:
  \(\mathrm{(\sqrt[3]{x})^3 = 3^3}\)

• Since cubing and cube root are inverse operations:
  \(\mathrm{x = 27}\)

4. Verify the answer

• Check: \(\mathrm{g(27) = 5\sqrt[3]{27} = 5(3) = 15}\) ✓

Answer: C (27)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors, particularly when cubing 3, sometimes calculating \(\mathrm{3^3 = 9}\) instead of 27.

This confusion between squaring (\(\mathrm{3^2 = 9}\)) and cubing (\(\mathrm{3^3 = 27}\)) leads them to select Choice B (9).

Second Most Common Error:

Incomplete SIMPLIFY process: Students correctly find \(\mathrm{\sqrt[3]{x} = 3}\) but stop there, thinking 3 is the final answer rather than recognizing they need to find x itself.

This premature stopping leads them to select Choice A (3).

The Bottom Line:

This problem tests whether students can systematically work through inverse operations and complete multi-step algebraic solutions. The key is recognizing that finding \(\mathrm{\sqrt[3]{x} = 3}\) is an intermediate step, not the final answer.