= Solution
Use $[\eta]=ML^{-1}T^{-1}$ and $[dp/dx]=ML^{-2}T^{-2}$. Equating <mass>, length and time exponents gives
$$
\alpha+\gamma=0,\qquad-\alpha+\beta-2\gamma=3,\qquad-\alpha-2\gamma=-1,
$$
hence
$$
\boxed{\alpha=-1,\quad\beta=4,\quad\gamma=1.}
$$
<Dimensional analysis> leaves $k$ undetermined. If positive $Q$ denotes flow along increasing $x$, the signed pressure derivative is negative and its minus sign belongs in the proportionality constant; alternatively use the pressure-drop magnitude.
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