Let $x,y$ be real numbers such that $xy=1$. Let $T$ and $t$ be the largest and the smallest values of the expression:

If $T+t$ can be expressed in the form $\frac{m}{n}$ where $m,n$ are nonzero integers with $\text{GCD}(m,n)=1$, find the value of $m+n$.

Step 1: Use the algebraic identity:

Since $xy=1$, we have $(x+y{)}^2=(x-y{)}^2+4$.

Step 2: Substitute $(x+y{)}^2$ into the expression:

Step 3: Let $u=x-y$. Since $x,y$ can be any real numbers satisfying $xy=1$, $u=x-1/x$ can take any real value. We define the function:

Step 4: To find the range of $f(u)$, set $f(u)=k$ and form a quadratic equation:

Step 5: Since $u$ is real, the discriminant of this quadratic equation must be non-negative:

Step 6: The roots of $7k^2-18k+7=0$ represent the minimum $t$ and maximum $T$ values of $k$:

Step 7: Calculate $T+t$:

Comparing to $\frac{m}{n}$, we get $m=18$ and $n=7$ (which are coprime).

Step 8: Finally, find $m+n$: