To solve the multiplication of the given expressions and simplify it to its simplest form, follow these steps:
1. Write down the expressions:
[tex]\[
\frac{45 y}{y^2 - 4} \cdot \frac{y^2 + 2y}{18 y^2}
\][/tex]
2. Factorize the denominators and numerators where possible:
- The first expression [tex]\( \frac{45 y}{y^2 - 4} \)[/tex]:
[tex]\[
y^2 - 4 \text{ can be factored as } (y - 2)(y + 2)
\][/tex]
So, the expression becomes:
[tex]\[
\frac{45 y}{(y - 2)(y + 2)}
\][/tex]
- The second expression [tex]\( \frac{y^2 + 2y}{18 y^2} \)[/tex]:
[tex]\[
y^2 + 2y \text{ can be factored as } y(y + 2)
\][/tex]
So, the expression becomes:
[tex]\[
\frac{y(y + 2)}{18 y^2}
\][/tex]
3. Substitute the factored forms into the original multiplication:
[tex]\[
\frac{45 y}{(y - 2)(y + 2)} \cdot \frac{y(y + 2)}{18 y^2}
\][/tex]
4. Multiply the numerators together and the denominators together:
Numerator:
[tex]\[
45 y \cdot y (y + 2) = 45 y^2 (y + 2)
\][/tex]
Denominator:
[tex]\[
(y - 2)(y + 2) \cdot 18 y^2 = 18 y^2 (y - 2)(y + 2)
\][/tex]
So, the expression becomes:
[tex]\[
\frac{45 y^2 (y + 2)}{18 y^2 (y - 2)(y + 2)}
\][/tex]
5. Cancel out common factors in the numerator and denominator:
- The [tex]\( y + 2 \)[/tex] terms in the numerator and denominator cancel out.
- The [tex]\( y^2 \)[/tex] terms in the numerator and denominator cancel out.
- Simplify [tex]\(\frac{45}{18}\)[/tex]:
[tex]\[
\frac{45}{18} = \frac{45 \div 9}{18 \div 9} = \frac{5}{2}
\][/tex]
6. Simplify the remaining expression:
After canceling out the common factors, the expression simplifies to:
[tex]\[
\frac{5}{2(y - 2)}
\][/tex]
7. Write the final simplified product:
[tex]\[
\frac{5}{2(y - 2)}
\][/tex]
Thus, the product of [tex]\( \frac{45 y}{y^2 - 4} \cdot \frac{y^2 + 2y}{18 y^2} \)[/tex] in its simplest form is:
[tex]\[
\frac{5}{2(y - 2)}
\][/tex]
