The function h is defined by \(\mathrm{h(t) = 8t^{3/2} + 5}\). What is the value of \(\mathrm{h(9)}\)? 113 216 221...

1. TRANSLATE the problem information

• Given: \(\mathrm{h(t) = 8t^{3/2} + 5}\)
• Find: \(\mathrm{h(9)}\)
• What this means: Substitute \(\mathrm{t = 9}\) into the function

2. TRANSLATE to set up the substitution

• Replace every t with 9:

\(\mathrm{h(9) = 8(9)^{3/2} + 5}\)

3. SIMPLIFY the fractional exponent

• Use the rule \(\mathrm{a^{m/n} = (\sqrt[n]{a})^m}\)
• So \(\mathrm{9^{3/2} = (\sqrt{9})^3}\)
• Since \(\mathrm{\sqrt{9} = 3}\): \(\mathrm{(\sqrt{9})^3 = 3^3 = 27}\)

4. SIMPLIFY the final calculation

• \(\mathrm{h(9) = 8(27) + 5}\)
• \(\mathrm{h(9) = 216 + 5 = 221}\)

Answer: C) 221

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak conceptual knowledge of fractional exponents: Students may not remember or correctly apply the rule \(\mathrm{a^{m/n} = (\sqrt[n]{a})^m}\). They might try to calculate \(\mathrm{9^{3/2}}\) incorrectly, such as treating it as \(\mathrm{9 \times 3/2 = 13.5}\), leading to \(\mathrm{h(9) = 8(13.5) + 5 = 113}\). This may lead them to select Choice A (113).

Second Most Common Error:

Poor SIMPLIFY execution: Students understand the fractional exponent rule but make arithmetic errors. For example, they might correctly find \(\mathrm{9^{3/2} = 27}\) but then calculate \(\mathrm{8 \times 27 = 206}\) instead of 216, giving \(\mathrm{h(9) = 206 + 5 = 211}\). Since this isn't an answer choice, this leads to confusion and guessing.

The Bottom Line:

The key challenge is correctly handling fractional exponents. Students need to remember that the denominator becomes the root and the numerator becomes the power, then execute the arithmetic carefully.