Which expression is equivalent to sqrt[4]{y^3}, where y gt 0?

1. TRANSLATE the radical expression to exponential form

• Given: \(\sqrt[4]{y^3}\) where \(\mathrm{y \gt 0}\)
• We need to use the conversion rule: \(\sqrt[n]{a^m} = a^{m/n}\)
• This rule lets us convert any radical expression to exponential form

2. INFER the values of n, m, and a from the expression

• Looking at \(\sqrt[4]{y^3}\):
  • The small 4 outside the radical symbol is the index: \(\mathrm{n = 4}\)
  • The exponent inside the radical is: \(\mathrm{m = 3}\)
  • The base is: \(\mathrm{a = y}\)

3. SIMPLIFY by applying the conversion rule

• Using \(\sqrt[n]{a^m} = a^{m/n}\) with our values:
• \(\sqrt[4]{y^3} = y^{m/n} = y^{3/4}\)

Answer: A. \(y^{3/4}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Process Skill - Weak INFER reasoning: Students mix up which number becomes the numerator versus denominator in the fractional exponent.

They might think: "The 4 is bigger, so it should go on top" or incorrectly remember the conversion rule as \(\sqrt[n]{a^m} = a^{n/m}\). This leads them to write \(y^{4/3}\) instead of \(y^{3/4}\).

This may lead them to select Choice B (\(y^{4/3}\))

Second Most Common Error:

Conceptual gap - Missing conversion rule knowledge: Students don't remember or understand the radical-to-exponential conversion rule, so they try other approaches like multiplying the numbers.

They might think: "4 times 3 equals 12, so the answer involves 12" and look for \(y^{12}\) among the choices.

This may lead them to select Choice C (\(y^{12}\))

The Bottom Line:

This problem tests whether students can correctly apply the radical-to-exponential conversion rule. The key challenge is keeping track of which number (the radical index or the inside exponent) becomes the denominator in the fractional exponent.