B1. Simplify the expression $\left(\frac{1}{2}-\frac{6}{a}+18 a^{-2}\right) \cdot \frac{2 a^{2}-4 a}{a-6}:\left(a-\frac{36}{a}\right)$.
B1. We simplify the expression in the parentheses and get $\frac{a^{2}-12 a+36}{2 a^{2}}$. We also simplify the expression in the second parentheses and get $\frac{a^{2}-36}{a}$. We convert the division into multiplication by the reciprocal, which is $\frac{a}{a^{2}-36}$. We factor the numerators and denominators of all fractions: $\frac{(a-6)(a+6)}{2 a^{2}} \cdot \frac{2 a(a-2)}{a-6} \cdot \frac{a}{(a-6)(a+6)}$, simplify, and get the solution $\frac{a-2}{a+6}$.

Simplification of the first parentheses $\frac{1}{2}-\frac{6}{a}+18 a^{-2}=\frac{a^{2}-12 a+36}{2 a^{2}} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$

![](https://cdn.mathpix.com/cropped/2024_06_07_ec35f4a613e63419eaedg-11.jpg?height=74&width=1687&top_left_y=551&top_left_x=230)

![](https://cdn.mathpix.com/cropped/2024_06_07_ec35f4a613e63419eaedg-11.jpg?height=71&width=1685&top_left_y=610&top_left_x=228)

Factoring the second fraction $\frac{2 a^{a}-4 a}{a-6}=\frac{2 a(a-2)}{a-6} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots 1$ point

Converting the division into a product $\frac{(a-6)(a+6)}{2 a^{2}} \cdot \frac{2 a(a-2)}{a-6} \cdot \frac{a}{(a-6)(a+6)} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots 1$ point

![](https://cdn.mathpix.com/cropped/2024_06_07_ec35f4a613e63419eaedg-11.jpg?height=68&width=1685&top_left_y=794&top_left_x=228)