Which expression is equivalent to sqrt[4]{x^(12)y^(8)}, where x and y are positive?x^(6)y^(4)x^(4)y^(2)x^(3)y^(2)x^(2)y^(3)x^(3)y^(4)

1. TRANSLATE the radical notation to exponential form

• Given: \(\sqrt[4]{x^{12}y^8}\) where x and y are positive
• The fourth root symbol \(\sqrt[4]{}\) means "to the 1/4 power"
• So: \(\sqrt[4]{x^{12}y^8} = (x^{12}y^8)^{1/4}\)

2. SIMPLIFY by distributing the exponent

• Use the rule: \((ab)^n = a^n \times b^n\)
• Apply this: \((x^{12}y^8)^{1/4} = (x^{12})^{1/4} \times (y^8)^{1/4}\)

3. SIMPLIFY each factor using the power rule

• Use the rule: \((a^m)^n = a^{mn}\)
• For x: \((x^{12})^{1/4} = x^{12 \times 1/4} = x^3\)
• For y: \((y^8)^{1/4} = y^{8 \times 1/4} = y^2\)

4. Combine the results

• \(x^3 \times y^2 = x^3y^2\)

Answer: C (\(x^3y^2\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make arithmetic errors when multiplying the exponents by 1/4.

They might calculate \(12 \times 1/4 = 4\) instead of 3, or \(8 \times 1/4 = 4\) instead of 2. These errors often happen when students rush through the fraction multiplication or confuse the operation (thinking addition instead of multiplication).

This may lead them to select Choice B (\(x^4y^2\)) or Choice E (\(x^3y^4\)) depending on which calculation they mess up.

Second Most Common Error:

Missing TRANSLATE skill: Students don't know how to convert radical notation to exponential form.

Without knowing that \(\sqrt[4]{a} = a^{1/4}\), they can't begin the systematic solution. They might try to guess patterns or apply incorrect "rules" they've invented, leading to confusion and random answer selection.

This leads to confusion and guessing among the available choices.

The Bottom Line:

This problem tests whether students can fluently work with two different notations for the same mathematical concept (radicals vs. exponents) and then systematically apply exponent rules without computational errors.