Let x be a positive real number. If sqrt[3]{x^(12)} * sqrt[4]{x^8} = x^k, what is the value of k? 8...

1. TRANSLATE the radical expressions to exponential form

• Given: \(\sqrt[3]{x^{12}} \cdot \sqrt[4]{x^{8}} = x^{k}\)
• Convert each radical using fractional exponents:
  • \(\sqrt[3]{x^{12}} = (x^{12})^{1/3}\)
  • \(\sqrt[4]{x^{8}} = (x^{8})^{1/4}\)

2. SIMPLIFY each exponential expression

• For the first term: \((x^{12})^{1/3} = x^{12 \times 1/3} = x^{12/3} = x^{4}\)
• For the second term: \((x^{8})^{1/4} = x^{8 \times 1/4} = x^{8/4} = x^{2}\)

3. SIMPLIFY the multiplication using exponent rules

• Now we have: \(x^{4} \cdot x^{2} = x^{4+2} = x^{6}\)
• Since this equals \(x^{k}\), we get: \(x^{k} = x^{6}\)
• Therefore: \(k = 6\)

Answer: D (6)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may try to simplify the expressions inside the radicals first, thinking \(\sqrt[3]{x^{12}}\) means they need to find what \(x^{12}\) equals, rather than converting to exponential form.

They might attempt operations like \(x^{12} \div 3 = x^{4}\) or \(x^{8} \div 4 = x^{2}\) without proper exponential reasoning, leading to confusion about how to proceed systematically.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly convert to exponential form but make arithmetic errors when computing the fractional exponents (getting \(12/3 = 3\) instead of 4, or \(8/4 = 3\) instead of 2) or incorrectly apply the multiplication rule (adding exponents wrong or forgetting the rule entirely).

This may lead them to select Choice A (8) if they incorrectly get \(x^{8}\), or Choice B (5) if they get confused and add \(12 + 8\) - some arbitrary adjustment.

The Bottom Line:

This problem tests whether students can fluently convert between radical and exponential notation, then systematically apply exponent rules - skills that require both conceptual understanding and careful arithmetic execution.