If x neq 0, which of the following expressions is equivalent to (sqrt(16x^4y^8))/x^3?

1. INFER the solution strategy

• We have a square root expression in the numerator and a variable term in the denominator
• Strategy: First simplify the square root, then handle the division

2. SIMPLIFY the square root in the numerator

• \(\sqrt{16\mathrm{x}^4\mathrm{y}^8} = \sqrt{16} \times \sqrt{\mathrm{x}^4} \times \sqrt{\mathrm{y}^8}\)
• \(\sqrt{16} = 4\)
• \(\sqrt{\mathrm{x}^4} = \mathrm{x}^2\) (since we're taking the square root of \(\mathrm{x}^4\))
• \(\sqrt{\mathrm{y}^8} = \mathrm{y}^4\) (since we're taking the square root of \(\mathrm{y}^8\))
• Result: \(4\mathrm{x}^2\mathrm{y}^4\)

3. SIMPLIFY the division

• Now we have: \(\frac{4\mathrm{x}^2\mathrm{y}^4}{\mathrm{x}^3}\)
• Separate the terms: \(4 \times \frac{\mathrm{x}^2}{\mathrm{x}^3} \times \mathrm{y}^4\)
• Apply division rule for powers: \(\frac{\mathrm{x}^2}{\mathrm{x}^3} = \mathrm{x}^{(2-3)} = \mathrm{x}^{-1}\)
• Result: \(4\mathrm{x}^{-1}\mathrm{y}^4\)

Answer: D. \(4\mathrm{x}^{-1}\mathrm{y}^4\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students incorrectly calculate the square root of the numerical coefficient or make errors with the variable exponents.

For example, they might calculate \(\sqrt{16} = 8\) instead of 4, or think \(\sqrt{\mathrm{x}^4} = \mathrm{x}^4\) instead of \(\mathrm{x}^2\). This leads to incorrect expressions like \(8\mathrm{x}^4\mathrm{y}^8/\mathrm{x}^3\), which doesn't match any answer choice, causing confusion and guessing.

Second Most Common Error:

Inadequate SIMPLIFY execution with negative exponents: Students correctly find \(\frac{4\mathrm{x}^2\mathrm{y}^4}{\mathrm{x}^3}\) but struggle with the final step of expressing the division as a negative exponent.

They might write the final answer as \(\frac{4\mathrm{y}^4}{\mathrm{x}}\) or leave it as a fraction, not recognizing this equals \(4\mathrm{x}^{-1}\mathrm{y}^4\). This may lead them to select Choice B (\(4\mathrm{xy}^4\)) if they mistakenly think \(1/\mathrm{x}\) equals \(\mathrm{x}^1\).

The Bottom Line:

This problem tests multiple algebraic skills in sequence - students must be solid on square root properties, power rules, and negative exponent notation to avoid getting derailed at any step.