Let's multiply and simplify the given fractions step-by-step:
[tex]\[ \frac{2y}{3x} \cdot \frac{9x^3y^2}{10y} \][/tex]
1. Multiply the numerators:
[tex]\[ (2y) \cdot (9x^3y^2) = 2 \cdot 9 \cdot y \cdot x^3 \cdot y^2 = 18x^3y^3 \][/tex]
2. Multiply the denominators:
[tex]\[ (3x) \cdot (10y) = 3 \cdot 10 \cdot x \cdot y = 30xy \][/tex]
Now we have:
[tex]\[ \frac{18x^3y^3}{30xy} \][/tex]
3. Simplify the fraction:
- Simplify the coefficients 18 and 30. Both 18 and 30 have a greatest common divisor of 6.
[tex]\[ \frac{18}{30} = \frac{18 \div 6}{30 \div 6} = \frac{3}{5} \][/tex]
- Simplify the variables:
- [tex]\( x^3 \)[/tex] in the numerator and [tex]\( x \)[/tex] in the denominator can be simplified:
[tex]\[ \frac{x^3}{x} = x^{3-1} = x^2 \][/tex]
- [tex]\( y^3 \)[/tex] in the numerator and [tex]\( y \)[/tex] in the denominator can be simplified:
[tex]\[ \frac{y^3}{y} = y^{3-1} = y^2 \][/tex]
So, after simplifying the variables, we get:
[tex]\[ \frac{3x^2y^2}{5} \][/tex]
Therefore, the simplified form of the product is:
[tex]\[ \boxed{\frac{3x^2y^2}{5}} \][/tex]
