Solve the Equation: Simplifying (x/y)^{-7} * (y/x) * (y/x)^{-2}

To solve this problem, we must simplify the expression $\left( \frac{x}{y} \right)^{-7} \cdot \frac{y}{x} \cdot \left( \frac{y}{x} \right)^{-2}$.

First, let's convert all negative exponents to positive using the rule $a^{-n} = \frac{1}{a^n}$:

• $\left( \frac{x}{y} \right)^{-7} = \left( \frac{y}{x} \right)^{7}$
• $\left( \frac{y}{x} \right)^{-2} = \left( \frac{x}{y} \right)^{2}$

The expression becomes:

$\left( \frac{y}{x} \right)^{7} \cdot \frac{y}{x} \cdot \left( \frac{x}{y} \right)^{2}$

Rewrite $\frac{y}{x}$ as $\left( \frac{y}{x} \right)^{1}$. The expression is now:

$\left( \frac{y}{x} \right)^{7} \cdot \left( \frac{y}{x} \right)^{1} \cdot \left( \frac{x}{y} \right)^{2}$

Let's combine the powers of the same base:

$\left( \frac{y}{x} \right)^{7+1} \cdot \left( \frac{x}{y} \right)^{2} = \left( \frac{y}{x} \right)^{8} \cdot \left( \frac{x}{y} \right)^{2}$

Now, apply the exponent rules again:

$\left( \frac{y}{x} \right)^{8} \cdot \left( \frac{x}{y} \right)^{2} = \left( \frac{y^{8} \cdot x^2}{x^8 \cdot y^2} \right)$

Simplify by using $y^{8} \cdot y^{-2} = y^{6}$ and $x^{2} \cdot x^{-8} = x^{-6}$:

$\frac{y^6}{x^6} = \left( \frac{y}{x} \right)^6$

Therefore, the simplified form of the expression is $\left( \frac{y}{x} \right)^6$, which corresponds to choice 1.