Answer
$x=6$.
Work Step by Step
$8^{-x+14}=(2^3)^{-x+14}=2^{-3x+42}$, $16^x=(2^4)^x=2^{4x}$ hence the equation becomes: \begin{align*}2^{-3x+42}=2^{4x}.\end{align*}
The base is same on the 2 sides on the equation (and it is not $1$ or $-1$), hence they will be equal if the exponents are equal. Hence $-3x+42=4x\\42=7x\\x=6$.
