Which expression is equivalent to sqrt[7]{x^9y^9}, where x and y are positive?

1. INFER the best approach

• Given: \(\sqrt[7]{x^9y^9}\)
• Strategy: Combine the terms under one radical first, then convert to exponential form for easier manipulation
• This approach avoids working with multiple separate radicals

2. SIMPLIFY by combining terms under the radical

• Use the reverse of the property \(a^m \cdot b^m = (ab)^m\):
• \(x^9y^9 = (xy)^9\)
• Therefore: \(\sqrt[7]{x^9y^9} = \sqrt[7]{(xy)^9}\)

3. SIMPLIFY by converting to exponential form

• Apply the property \(\sqrt[n]{a} = a^{1/n}\):
• \(\sqrt[7]{(xy)^9} = ((xy)^9)^{1/7}\)

4. SIMPLIFY using the power rule

• Apply \((a^m)^n = a^{mn}\):
• \(((xy)^9)^{1/7} = (xy)^{9 \times 1/7} = (xy)^{9/7}\)

Answer: B. \((xy)^{9/7}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students try to separate the radical immediately: \(\sqrt[7]{x^9y^9} = \sqrt[7]{x^9} \cdot \sqrt[7]{y^9} = x^{9/7} \cdot y^{9/7}\)

They correctly convert each part to exponential form but fail to recognize that the final step should combine these into \((xy)^{9/7}\). Instead, they leave the answer as a product of two separate exponential terms, which doesn't match any of the given choices. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(((xy)^9)^{1/7}\) but make an arithmetic error with the exponent calculation, thinking that \(9 \times (1/7) = 7/9\) instead of \(9/7\).

This may lead them to select Choice A. \((xy)^{7/9}\).

The Bottom Line:

This problem tests the strategic decision of when to combine terms versus when to separate them. The key insight is recognizing that combining first (using the reverse of exponent properties) creates a cleaner path to the exponential form than working with separated radicals.