Which of the following expressions is equivalent to the expression above?sqrt[3]{x^3y^6}

1. INFER the approach needed

• I have a cube root of a product: \(\sqrt[3]{\mathrm{x}^3\mathrm{y}^6}\)
• Key insight: I can use the radical property to separate this into simpler pieces

2. SIMPLIFY using radical properties

• Apply \(\sqrt[3]{\mathrm{ab}} = \sqrt[3]{\mathrm{a}} \cdot \sqrt[3]{\mathrm{b}}\):
  \(\sqrt[3]{\mathrm{x}^3\mathrm{y}^6} = \sqrt[3]{\mathrm{x}^3} \cdot \sqrt[3]{\mathrm{y}^6}\)

3. SIMPLIFY each cube root separately

• For \(\sqrt[3]{\mathrm{x}^3}\): The exponent rule gives us \(\mathrm{x}^{3/3} = \mathrm{x}^1 = \mathrm{x}\)
• For \(\sqrt[3]{\mathrm{y}^6}\): The exponent rule gives us \(\mathrm{y}^{6/3} = \mathrm{y}^2\)

4. Combine the results

• \(\sqrt[3]{\mathrm{x}^3\mathrm{y}^6} = \mathrm{x} \cdot \mathrm{y}^2 = \mathrm{xy}^2\)

Answer: B. \(\mathrm{xy}^2\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students incorrectly apply the exponent rule for \(\sqrt[3]{\mathrm{y}^6}\)

Many students think that since we're taking a cube root, \(\sqrt[3]{\mathrm{y}^6}\) should equal \(\mathrm{y}^3\). They incorrectly reason "6 divided by something gives 3, so the answer has \(\mathrm{y}^3\)." However, the correct rule is \(\mathrm{y}^{6/3} = \mathrm{y}^2\), not \(\mathrm{y}^3\).

This may lead them to select Choice D (\(\mathrm{xy}^3\))

Second Most Common Error:

Missing conceptual knowledge: Students don't remember the radical property \(\sqrt[3]{\mathrm{ab}} = \sqrt[3]{\mathrm{a}} \cdot \sqrt[3]{\mathrm{b}}\)

Without this property, students can't break apart the expression systematically. They might try to work with \(\sqrt[3]{\mathrm{x}^3\mathrm{y}^6}\) as one piece or guess based on pattern recognition, potentially forgetting about the x term entirely.

This may lead them to select Choice A (\(\mathrm{y}^2\)) by focusing only on the y portion.

The Bottom Line:

This problem tests whether students can systematically break down complex radical expressions using properties rather than trying to work with them as single units. The key is recognizing that radicals can be separated when dealing with products.