Question:The function g is defined by \(\mathrm{g(x) = 10 - x^{2/3}}\). What is the value of \(\mathrm{g(27)}\)?-81719

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{g(x) = 10 - x^{2/3}}\)
  • Need to find: \(\mathrm{g(27)}\)
• What this tells us: We need to substitute \(\mathrm{x = 27}\) into the function and evaluate

2. TRANSLATE the substitution

• Replace x with 27 in the function:
  \(\mathrm{g(27) = 10 - 27^{2/3}}\)
• Our task: Calculate \(\mathrm{27^{2/3}}\) and subtract from 10

3. SIMPLIFY the fractional exponent

• Key insight: \(\mathrm{27^{2/3} = (27^{1/3})^2}\)
• This means: find the cube root of 27, then square the result
• Step 1: Find \(\mathrm{\sqrt[3]{27} = 3}\) (since \(\mathrm{3 × 3 × 3 = 27}\))
• Step 2: Square the result: \(\mathrm{3^2 = 9}\)
• Therefore: \(\mathrm{27^{2/3} = 9}\)

4. SIMPLIFY the final calculation

• Substitute back into the function:
  \(\mathrm{g(27) = 10 - 9 = 1}\)

Answer: B. 1

Why Students Usually Falter on This Problem

Most Common Error Path:

Fractional exponent confusion: Students may not understand that \(\mathrm{x^{2/3}}\) means \(\mathrm{(\sqrt[3]{x})^2}\). Instead, they might try to calculate it as \(\mathrm{(x^2)^{1/3}}\) or even incorrectly as \(\mathrm{x^2/x^3}\).

For example, some students calculate \(\mathrm{27^2 = 729}\), then try to find \(\mathrm{\sqrt[3]{729} ≈ 8.99}\), leading to \(\mathrm{g(27) = 10 - 9 = 1}\) by coincidence, or they get confused and make errors that lead them to select Choice A (-8) or Choice D (19).

Second Most Common Error:

Weak SIMPLIFY execution: Students correctly identify that they need \(\mathrm{(\sqrt[3]{27})^2}\), but make arithmetic errors—either miscalculating \(\mathrm{\sqrt[3]{27}}\) as something other than 3, or incorrectly squaring the result.

This leads to incorrect values for \(\mathrm{27^{2/3}}\), causing them to select Choice C (7) if they get \(\mathrm{27^{2/3} = 3}\), or other incorrect choices based on their calculation errors.

The Bottom Line:

This problem tests whether students can correctly interpret and calculate fractional exponents. The key is recognizing that the fraction 2/3 in the exponent tells you the order of operations: cube root first (denominator), then square (numerator).