2. Simplify the expression $\left(\frac{x^{2}}{y^{2}}+\frac{y^{2}}{x^{2}}-1\right):\left(\frac{x^{3}}{y^{3}}+\frac{y^{3}}{x^{3}}\right)$ and calculate its exact value for $x=\sqrt{33}$ and $y=\sqrt{77}$.

(6 points)
2. We simplify the expressions in parentheses or expand the fractions to common denominators: the first parenthesis $\frac{x^{4}+y^{4}-x^{2} y^{2}}{x^{2} y^{2}}$ and the second parenthesis $\frac{x^{6}+y^{6}}{x^{3} y^{3}}$. We convert the expression from division to multiplication. We use the formula for the sum of cubes and factor the second denominator $x^{6}+y^{6}=\left(x^{2}+y^{2}\right)\left(x^{4}-x^{2} y^{2}+y^{4}\right)$. We simplify the fractions and substitute the values for $x$ and $y$ into the simplified expression $\frac{x y}{x^{2}+y^{2}}$. We partially root the numerator $\sqrt{33 \cdot 77}=11 \sqrt{21}$, simplify the fraction, and obtain the result $\frac{\sqrt{21}}{10}$.

![](https://cdn.mathpix.com/cropped/2024_06_07_7bf3e9b12130e254269ag-06.jpg?height=86&width=1639&top_left_y=2404&top_left_x=206)

![](https://cdn.mathpix.com/cropped/2024_06_07_7bf3e9b12130e254269ag-06.jpg?height=77&width=1639&top_left_y=2480&top_left_x=206)

![](https://cdn.mathpix.com/cropped/2024_06_07_7bf3e9b12130e254269ag-06.jpg?height=74&width=1636&top_left_y=2539&top_left_x=207)

Simplified expression $\frac{x y}{x^{2}+y^{2}}$..............................................................................................................................

![](https://cdn.mathpix.com/cropped/2024_06_07_7bf3e9b12130e254269ag-06.jpg?height=57&width=1636&top_left_y=2650&top_left_x=210)

Result $\frac{\sqrt{21}}{10}$