To simplify the expression [tex]\(\frac{x^3 + 3y^2}{xy}\)[/tex], follow these steps:
1. Distribute the Denominator:
You can split the numerator by distributing the denominator across each term in the numerator:
[tex]\[
\frac{x^3 + 3y^2}{xy} = \frac{x^3}{xy} + \frac{3y^2}{xy}
\][/tex]
2. Simplify Each Fraction:
Simplify each term individually:
- For the first term, [tex]\(\frac{x^3}{xy}\)[/tex]:
[tex]\[
\frac{x^3}{xy} = \frac{x^3}{x \cdot y} = \frac{x^3}{x \cdot y} = x^{3-1} \cdot \frac{1}{y} = \frac{x^2}{y}
\][/tex]
- For the second term, [tex]\(\frac{3y^2}{xy}\)[/tex]:
[tex]\[
\frac{3y^2}{xy} = 3 \cdot \frac{y^2}{xy} = 3 \cdot \frac{y^2}{y \cdot x} = 3 \cdot \frac{y^2}{y \cdot x} = 3 \cdot \frac{y^{2-1}}{x} = \frac{3y}{x}
\][/tex]
3. Combine the Simplified Terms:
Putting the simplified terms together, you get:
[tex]\[
\frac{x^2}{y} + \frac{3y}{x}
\][/tex]
So, the simplified form of the given expression [tex]\(\frac{x^3 + 3y^2}{xy}\)[/tex] is:
[tex]\[
\frac{x^2}{y} + \frac{3y}{x}
\][/tex]
