Answer
$2-\displaystyle \frac{1}{2}\log_{8}(x+1)$
Work Step by Step
$\displaystyle \log_{8}(\frac{64}{\sqrt{x+1}})=$
...apply the Quotient Rule: $\displaystyle \quad \log_{b}(\frac{M}{N})=\log_{b}\mathrm{M}-\log_{b}\mathrm{N}$
$\log_{8}64-\log_{8}\sqrt{x+1}$
... write $64$ as $8^{2},\quad\sqrt{x+1}=(x+1)^{1/2}$
$=\log_{8}8^{2}-\log_{8}(x+1)^{1/2}$
$\quad $...apply the Power Rule: $\quad \log_{b}(M^{p})=p\cdot\log_{b}\mathrm{M}$
$=2\displaystyle \log_{8}8-\frac{1}{2}\log_{8}(x+1)$
$\quad $...apply the basic property: $\log_{b}b=1$
$=2-\displaystyle \frac{1}{2}\log_{8}(x+1)$