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:

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<reasoning>

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.
</reasoning>

<answer>

Question 1

<mcq-correct>(A) \(\frac{x^{\frac{2}{3}}y}{x^{\frac{2}{3}}y^{\frac{1}{3}}}\)</mcq-correct>
<mcq-option>(B) \(\frac{x^{\frac{7}{3}}y^{\frac{10}{3}}}{x^{\frac{7}{3}}y^{\frac{4}{3}}}\)</mcq-option>
<mcq-option>(C) \(\frac{x^{\frac{5}{3}}y^{\frac{8}{3}}}{x^{\frac{5}{3}}y^{\frac{2}{3}}}\)</mcq-option>

Question 2

<mcq-option>(A) \(y^{\frac{10}{3}}\)</mcq-option>
<mcq-correct>(B) \(y^{\frac{2}{3}}\)</mcq-correct>
<mcq-option>(C) \(y^2\)</mcq-option>
</answer>

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"Algebra",
"Exponent Rules"
]
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</post_analysis>

Answer:

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<reasoning>

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.
</reasoning>

<answer>

Question 1

<mcq-correct>(A) \(\frac{x^{\frac{2}{3}}y}{x^{\frac{2}{3}}y^{\frac{1}{3}}}\)</mcq-correct>
<mcq-option>(B) \(\frac{x^{\frac{7}{3}}y^{\frac{10}{3}}}{x^{\frac{7}{3}}y^{\frac{4}{3}}}\)</mcq-option>
<mcq-option>(C) \(\frac{x^{\frac{5}{3}}y^{\frac{8}{3}}}{x^{\frac{5}{3}}y^{\frac{2}{3}}}\)</mcq-option>

Question 2

<mcq-option>(A) \(y^{\frac{10}{3}}\)</mcq-option>
<mcq-correct>(B) \(y^{\frac{2}{3}}\)</mcq-correct>
<mcq-option>(C) \(y^2\)</mcq-option>
</answer>

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"question_type": "Multiple Choice",
"knowledge_point": [
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"Algebra",
"Exponent Rules"
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