## Task B-1.3.

For real numbers $x, y$, if $x^{-1}-y^{-1}=4$ and $x y=\frac{1}{4}$, what is $x^{-4}+y^{-4}$?
## Solution.

By squaring the equality $x^{-1}-y^{-1}=4$, we get $x^{-2}-2 x^{-1} y^{-1}+y^{-2}=16$.

If we substitute $x^{-1} y^{-1}=4$ into the obtained equality, we conclude that $x^{-2}+y^{-2}=24$.

Squaring the previous equality, we find that $x^{-4}+2 x^{-2} y^{-2}+y^{-4}=576$.

Furthermore, since $x^{-1} y^{-1}=4$ implies $x^{-2} y^{-2}=16$, substituting into

$x^{-4}+2 x^{-2} y^{-2}+y^{-4}=576$, we finally get that $x^{-4}+y^{-4}=544$.