$1.16 \frac{x^{2}+4 x-5+(x-5) \sqrt{x^{2}-1}}{x^{2}-4 x-5+(x+5) \sqrt{x^{2}-1}}, x>1$.

$\mathbf{1 . 1 7}$

$$
9 b^{\frac{4}{3}}-\frac{a^{\frac{3}{2}}}{b^{2}}
$$
1.16 Let us denote the given expression by $f(x)$. Further, using formula (1.15), we factorize the quadratic trinomials in the numerator and the denominator of the fraction:

$$
f(x)=\frac{(x+5)(x-1)+(x-5) \sqrt{x^{2}-1}}{(x-5)(x+1)+(x+5) \sqrt{x^{2}-1}}
$$

Since $x>1$, by relation (1.21) we have

$$
\begin{aligned}
& x-1=\sqrt{(x-1)^{2}} \text { and } x+1=\sqrt{(x+1)^{2}} \cdot \text { Therefore, } \\
& f(x)=\frac{\sqrt{x-1}((x+5) \sqrt{x-1}+(x-5) \sqrt{x+1})}{\sqrt{x+1}((x-5) \sqrt{x+1}+(x+5) \sqrt{x-1})}
\end{aligned}
$$

from which, after simplification, we obtain

$$
f(x)=\sqrt{\frac{x-1}{x+1}}
$$

Answer: $\sqrt{\frac{x-1}{x+1}}$.