Solve: xy-(156÷(13/xy)+36÷(8xy)) - Complex Algebraic Expression

To solve this problem, we'll follow these steps:

• Step 1: Simplify the fractions $\frac{13}{xy}$ and $\frac{36}{8 \cdot xy}$.
• Step 2: Substitute these simplified fractions into the larger expression.
• Step 3: Simplify the entire expression by performing arithmetic operations and combining like terms.

Now, let's work through each step:
Step 1: Simplify the fractions:
The first fraction is $\frac{13}{xy}$, and the second fraction is $\frac{36}{8 \cdot xy} = \frac{36}{8xy} = \frac{9}{2xy}$ after simplifying $\frac{36}{8}$ to $\frac{9}{2}$. Thus, the fractions are $\frac{13}{xy}$ and $\frac{9}{2xy}$.

Step 2: Substitute into the expression:
We substitute back into the expression:
$xy - \left( \frac{156}{\frac{13}{xy}} + \frac{36}{8 \cdot xy} \right)$.
This becomes:
$xy - \left( 156 \cdot \frac{xy}{13} + 9 \cdot \frac{1}{2xy} \right)$ by converting division to multiplication by reciprocals.

Step 3: Simplify further:
Simplify $156 \cdot \frac{xy}{13}$ to $12xy$ (since $156 \div 13 = 12$) and $9 \cdot \frac{1}{2xy}$ to $\frac{9}{2xy}$.
Now the expression inside the brackets becomes $12xy + \frac{9}{2xy}$.

Combine with the outer term:
$xy - \left( 12xy + \frac{9}{2xy} \right) = xy - 12xy - \frac{9}{2xy}$.
Thus, the expression simplifies to:
$-11xy - \frac{9}{2xy} = -11xy - 4\frac{1}{2}\frac{1}{xy}$ after converting $\frac{9}{2xy}$ to $4\frac{1}{2}\frac{1}{xy}$.

Therefore, the solution to the problem is $-11xy - 4\frac{1}{2}\frac{1}{xy}$.