The function h is defined by \(\mathrm{h(x) = k \cdot x^{3/2}}\), where k is a positive constant. If \(\mathrm{h(4) =...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{h(x) = k \cdot x^{3/2}}\) where \(\mathrm{k}\) is a positive constant
  • \(\mathrm{h(4) = 96}\)
• What this tells us: We can use the known value \(\mathrm{h(4) = 96}\) to find \(\mathrm{k}\), then use \(\mathrm{k}\) to calculate \(\mathrm{h(9)}\)

2. TRANSLATE the given condition into an equation

• Since \(\mathrm{h(4) = 96}\) and \(\mathrm{h(x) = k \cdot x^{3/2}}\):
  \(\mathrm{k \cdot 4^{3/2} = 96}\)

3. SIMPLIFY to find the constant k

• Calculate \(\mathrm{4^{3/2}}\):
  \(\mathrm{4^{3/2} = (4^{1/2})^3}\)
  \(\mathrm{= (\sqrt{4})^3}\)
  \(\mathrm{= 2^3}\)
  \(\mathrm{= 8}\)
• Substitute back: \(\mathrm{k \cdot 8 = 96}\)
• Solve for \(\mathrm{k}\): \(\mathrm{k = 96 \div 8 = 12}\)

4. SIMPLIFY to find h(9)

• Now we know \(\mathrm{k = 12}\), so: \(\mathrm{h(9) = 12 \cdot 9^{3/2}}\)
• Calculate \(\mathrm{9^{3/2}}\):
  \(\mathrm{9^{3/2} = (9^{1/2})^3}\)
  \(\mathrm{= (\sqrt{9})^3}\)
  \(\mathrm{= 3^3}\)
  \(\mathrm{= 27}\)
• Final calculation: \(\mathrm{h(9) = 12 \cdot 27 = 324}\)

Answer: D) 324

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Misunderstanding fractional exponents

Many students incorrectly interpret \(\mathrm{x^{3/2}}\) as "3 divided by 2 times x" or confuse it with other operations. For example, they might calculate:

• \(\mathrm{4^{3/2}}\) as \(\mathrm{4 \times (3/2) = 6}\), leading to \(\mathrm{k = 96 \div 6 = 16}\)
• Then \(\mathrm{h(9) = 16 \times 6 = 96}\), which isn't among the answer choices

This leads to confusion and guessing.

Second Most Common Error:

Incomplete SIMPLIFY execution: Correct method but calculation errors

Students understand that \(\mathrm{4^{3/2} = (\sqrt{4})^3}\) but make arithmetic mistakes:

• Calculate \(\mathrm{(\sqrt{4})^3 = 2^2 = 4}\) instead of \(\mathrm{2^3 = 8}\)
• This gives \(\mathrm{k = 96 \div 4 = 24}\)
• Then \(\mathrm{h(9) = 24 \times 27 = 648}\), which is far from any answer choice

This causes them to get stuck and guess among the provided options.

The Bottom Line:

Success on this problem hinges on correctly handling fractional exponents. The key insight is recognizing that \(\mathrm{x^{3/2}}\) means "take the square root, then cube the result" - a two-step process that many students rush through or misunderstand.