If x gt 0, for what positive integer k does the expression sqrt[k]{x^(72)} equal x^(3/2)?

1. TRANSLATE the root expression into exponential form

• Given information:
  • \(\sqrt[k]{x^{72}} = x^{3/2}\)
  • \(x \gt 0\) and k is a positive integer

• TRANSLATE the k-th root: \(\sqrt[k]{x^{72}} = x^{72/k}\)

2. INFER the solving strategy

• Since we have \(x^{72/k} = x^{3/2}\), and both sides have the same base \(x \gt 0\)
• Key insight: When exponential expressions with the same positive base are equal, their exponents must be equal
• This means: \(\frac{72}{k} = \frac{3}{2}\)

3. SIMPLIFY the equation to find k

• Cross-multiply: \(72 \times 2 = 3k\)
• Calculate: \(144 = 3k\)
• Divide both sides by 3: \(k = \frac{144}{3} = 48\)

Answer: C (48)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students struggle to convert the root notation \(\sqrt[k]{x^{72}}\) into exponential form \(x^{72/k}\). They might think the k-th root means dividing by k in a different way, or they might try to work directly with the radical without converting to exponents.

This leads to confusion about how to set up the equation and they end up guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\frac{72}{k} = \frac{3}{2}\) but make arithmetic errors during cross-multiplication. They might calculate \(72 \times 2 = 3k\) as \(144 = 3k\) but then incorrectly compute \(k = 144 \div 3\), getting \(k = 36\) instead of \(k = 48\).

This may lead them to select Choice B (36).

The Bottom Line:

This problem tests your ability to work fluently between radical and exponential notation. The key breakthrough is recognizing that roots are just fractional exponents, which allows you to use the fundamental principle that equal bases require equal exponents.