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What is [tex]\frac{\left(x^2 y^3\right)^{\frac{1}{3}}}{\sqrt[3]{x^2 y}}[/tex] in exponential form?

A. [tex]\frac{x^{\frac{2}{3}} y}{x^{\frac{2}{3}} y^{\frac{1}{3}}}[/tex]

B. [tex]\frac{x^{\frac{7}{3}} y^{\frac{10}{3}}}{x^{\frac{7}{3}} y^{\frac{4}{3}}}[/tex]

C. [tex]x \frac{x^{\frac{5}{3}} y^{\frac{8}{3}}}{x^{\frac{5}{3}} y^{\frac{2}{3}}}[/tex]



Answer :

GinnyAnswer
Certainly! Let's break down and simplify each part of the expression step by step.

1. Simplify the first expression:
[tex]\[ \frac{\left(x^2 y^3\right)^{\frac{1}{3}}}{\sqrt[3]{x^2 y}} \][/tex]

- Simplify the numerator [tex]\((x^2 y^3)^{\frac{1}{3}}\)[/tex]:
[tex]\[ (x^2 y^3)^{\frac{1}{3}} = x^{2 \cdot \frac{1}{3}} y^{3 \cdot \frac{1}{3}} = x^{\frac{2}{3}} y \][/tex]

- Simplify the denominator [tex]\(\sqrt[3]{x^2 y}\)[/tex]:
[tex]\[ \sqrt[3]{x^2 y} = (x^2 y)^{\frac{1}{3}} = x^{2 \cdot \frac{1}{3}} y^{\frac{1}{3}} = x^{\frac{2}{3}} y^{\frac{1}{3}} \][/tex]

- Combine the simplified parts:
[tex]\[ \frac{x^{\frac{2}{3}} y}{x^{\frac{2}{3}} y^{\frac{1}{3}}} \][/tex]

- We can cancel out [tex]\(x^{\frac{2}{3}}\)[/tex]:
[tex]\[ \frac{y}{y^{\frac{1}{3}}} = y^{1 - \frac{1}{3}} = y^{\frac{2}{3}} \][/tex]

2. Simplify the second expression:
[tex]\[ \frac{x^{\frac{7}{3}} y^{\frac{10}{3}}}{x^{\frac{7}{3}} y^{\frac{4}{3}}} \][/tex]

- We can cancel out [tex]\(x^{\frac{7}{3}}\)[/tex]:
[tex]\[ \frac{y^{\frac{10}{3}}}{y^{\frac{4}{3}}} = y^{\frac{10}{3} - \frac{4}{3}} = y^{\frac{6}{3}} = y^2 \][/tex]

The simplified forms of the given expressions are thus:
1. [tex]\(\frac{\left(x^2 y^3\right)^{\frac{1}{3}}}{\sqrt[3]{x^2 y}}\)[/tex] simplifies to [tex]\(y^{\frac{2}{3}}\)[/tex]
2. [tex]\(\frac{x^{\frac{7}{3}} y^{\frac{10}{3}}}{x^{\frac{7}{3}} y^{\frac{4}{3}}}\)[/tex] simplifies to [tex]\(y^2\)[/tex]

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