Question

what is \frac{(x^2 y^3)^{\frac{1}{3}}}{\sqrt3{x^2 y}} in exponential form?

• \frac{x^{\frac{2}{3}} y}{x^{\frac{2}{3}} y^{\frac{1}{3}}}
• \frac{x^{\frac{7}{3}} y^{\frac{10}{3}}}{x^{\frac{7}{3}} y^{\frac{4}{3}}}
• \frac{x^{\frac{5}{3}} y^{\frac{8}{3}}}{x^{\frac{5}{3}} y^{\frac{2}{3}}}

what is \frac{x^{\frac{2}{3}} y}{x^{\frac{2}{3}} y^{\frac{1}{3}}} in simplest exponential form?

• y^{\frac{10}{3}}
• y^{\frac{2}{3}}
• y^2

Explanation:

Convert the numerator to exponential form

Using the Radical to Exponential Form and Exponent Rules knowledge points

$$ (x^2 y^3)^{\frac{1}{3}} = (x^2)^{\frac{1}{3}} (y^3)^{\frac{1}{3}} = x^{\frac{2}{3}} y^1 $$

Convert the denominator to exponential form

Using the Radical to Exponential Form and Exponent Rules knowledge points

$$ \sqrt[3]{x^2 y} = (x^2 y)^{\frac{1}{3}} = (x^2)^{\frac{1}{3}} y^{\frac{1}{3}} = x^{\frac{2}{3}} y^{\frac{1}{3}} $$

Combine the numerator and denominator

Using the Algebraic Simplification knowledge point

$$ \frac{x^{\frac{2}{3}} y^1}{x^{\frac{2}{3}} y^{\frac{1}{3}}} $$

This matches the first option of the first question.

Simplify the combined expression

Using the Exponent Rules and Algebraic Simplification knowledge points

$$ \frac{x^{\frac{2}{3}} y^1}{x^{\frac{2}{3}} y^{\frac{1}{3}}} = x^{\frac{2}{3} - \frac{2}{3}} y^{1 - \frac{1}{3}} = x^0 y^{\frac{2}{3}} = y^{\frac{2}{3}} $$

This matches the second option of the second question.

Answer:

Question 1

• (A) \(\frac{x^{\frac{2}{3}}y}{x^{\frac{2}{3}}y^{\frac{1}{3}}}\) (Correct answer)
• (B) \(\frac{x^{\frac{7}{3}}y^{\frac{10}{3}}}{x^{\frac{7}{3}}y^{\frac{4}{3}}}\)
• (C) \(\frac{x^{\frac{5}{3}}y^{\frac{8}{3}}}{x^{\frac{5}{3}}y^{\frac{2}{3}}}\)

Question 2

• (A) \(y^{\frac{10}{3}}\)
• (B) \(y^{\frac{2}{3}}\) (Correct answer)
• (C) \(y^2\)