<s>[INST] What is the value of $x*y$? 
Given that $$\left(\frac{x}{y}\right)^{-2} + \left(\frac{y}{x}\right)^{-2} = \frac{10}{3}$$
  find the value of $x*y$.

My question is, can we calculate the value of $x*y$ or not? If yes, then how? If not, then why?
 [/INST] Let $u = (\frac{x}{y})^2$.
Then, your equation becomes: $\frac{1}{u} + u = \frac{10}{3}$.
Multiplying both sides by u, we get:
$1 + u^2 = \frac{10}{3} \cdot u$
Now, we have a quadratic equation to solve. Subtract $\frac{10}{3} \cdot u$ from both sides:
$1 + u^2 - \frac{10}{3} \cdot u = 0$
For look's sake, let's put our new polynomial in standard form:
$u^2 - \frac{10}{3} \cdot u + 1 = 0$
Now, solve for u. Notice that there are two solutions. So, because we don't have any indication of which u to use, we can only say that y^2 is either approximately $\frac{x}{1.7676} $ or approximately $\frac{x}{0.5657}.$ Therefore, we cannot calculate xy.
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