To simplify the expression [tex]\(\frac{9 x^3}{y} \cdot \frac{2 y^3}{15 x^5 y}\)[/tex], follow the steps below:
1. Write the expression as a single fraction:
[tex]\[
\frac{9 x^3}{y} \cdot \frac{2 y^3}{15 x^5 y} = \frac{9 x^3 \cdot 2 y^3}{y \cdot 15 x^5 y}
\][/tex]
2. Multiply the numerators and the denominators:
- The numerator is [tex]\(9 x^3 \cdot 2 y^3 = 18 x^3 y^3\)[/tex].
- The denominator is [tex]\(y \cdot 15 x^5 y = 15 x^5 y^2\)[/tex].
So the fraction now looks like:
[tex]\[
\frac{18 x^3 y^3}{15 x^5 y^2}
\][/tex]
3. Simplify the fraction by canceling common factors in the numerator and denominator:
- For the constants, [tex]\(\frac{18}{15}\)[/tex] simplifies to [tex]\(\frac{6}{5}\)[/tex] because both numbers can be divided by 3.
- For the [tex]\(x\)[/tex] terms, [tex]\(\frac{x^3}{x^5} = \frac{1}{x^2}\)[/tex] because [tex]\(x^3 - x^5 = x^{-2}\)[/tex].
- For the [tex]\(y\)[/tex] terms, [tex]\(\frac{y^3}{y^2} = y\)[/tex] because [tex]\(y^3 - y^2 = y^{1}\)[/tex].
Combining these simplifications, we have:
[tex]\[
\frac{6 y}{5 x^2}
\][/tex]
So, the simplified form of the given expression is:
[tex]\[
\frac{6 y}{5 x^2}
\][/tex]
