MathNet 2017-part-04 — 176 problems (EN–VI) Question 88

Suppose that $x$ and $y$ are nonzero real numbers such that $\left(x+\frac{1}{y}\right)\left(y+\frac{1}{x}\right)=7$. Find the value of $\left(x^{2}+\frac{1}{y^{2}}\right)\left(y^{2}+\frac{1}{x^{2}}\right)$.