The function g is defined by \(\mathrm{g(x) = 8x^{-1/3}}\). What is the value of \(\mathrm{g(64)}\)? 1/4 1/2 2 32...

1. TRANSLATE the problem information

• Given: \(\mathrm{g(x) = 8x^{-1/3}}\)
• Find: \(\mathrm{g(64)}\)
• This means substitute \(\mathrm{x = 64}\) into the function

2. TRANSLATE by substitution

• \(\mathrm{g(64) = 8(64)^{-1/3}}\)
• Now we need to evaluate this expression with a negative fractional exponent

3. INFER the approach for negative fractional exponents

• The exponent \(\mathrm{-1/3}\) combines two rules we need to apply
• Negative exponents create reciprocals: \(\mathrm{x^{-n} = \frac{1}{x^n}}\)
• Fractional exponents create roots: \(\mathrm{x^{1/n} = \sqrt[n]{x}}\)
• Strategy: First handle the negative, then the fractional part

4. SIMPLIFY using exponent rules

• \(\mathrm{(64)^{-1/3} = \frac{1}{(64)^{1/3}}}\)
• \(\mathrm{(64)^{1/3}}\) means the cube root of 64
• Since \(\mathrm{4 \times 4 \times 4 = 64}\), we have \(\mathrm{(64)^{1/3} = 4}\)
• Therefore: \(\mathrm{(64)^{-1/3} = \frac{1}{4}}\)

5. SIMPLIFY the final calculation

• \(\mathrm{g(64) = 8 \times \frac{1}{4}}\)
• \(\mathrm{g(64) = \frac{8}{4} = 2}\)

Answer: C. 2

Why Students Usually Falter on This Problem

Most Common Error Path:

Incomplete SIMPLIFY execution: Students correctly find \(\mathrm{(64)^{-1/3} = \frac{1}{4}}\) but stop there, forgetting that the original function has a coefficient of 8. They see the \(\mathrm{\frac{1}{4}}\) result and immediately select Choice A \(\mathrm{(\frac{1}{4})}\) without completing the multiplication step.

Second Most Common Error:

Conceptual confusion about negative exponents: Students ignore or misapply the negative sign in the exponent, treating \(\mathrm{(64)^{-1/3}}\) as if it were \(\mathrm{(64)^{1/3} = 4}\). Then they calculate \(\mathrm{g(64) = 8 \times 4 = 32}\), leading them to select Choice D (32).

The Bottom Line:

This problem requires systematic application of multiple exponent rules in sequence. Students must resist the urge to stop at intermediate results and must carefully track negative signs throughout their calculations.