The power P delivered by a machine varies with distance x and time...

From the principle of homogeneity of dimensions, the quantity $\beta$ must have the same dimensions as $t^2$.

$[\beta] = [\mathrm{T}^2]$

The dimensional formula for power $P$ is $[\mathrm{M} \mathrm{L}^2 \mathrm{T}^{-3}]$ and for distance $x$ is $[\mathrm{L}]$.

Using the given equation $P = \dfrac{\alpha \sqrt{x}}{\beta + t^2}$, the dimensions of $\alpha$ are:

$[\alpha] = \dfrac{[P][\beta + t^2]}{[\sqrt{x}]}$

$[\alpha] = \dfrac{[\mathrm{M} \mathrm{L}^2 \mathrm{T}^{-3}][\mathrm{T}^2]}{[\mathrm{L}^{1/2}]} = [\mathrm{M} \mathrm{L}^{3/2} \mathrm{T}^{-1}]$

The dimensional formula for $\dfrac{\alpha^2}{\beta}$ is:

\left[\dfrac{\alpha^2}{\beta}\right] = \dfrac{[\mathrm{M} \mathrm{L}^{3/2} \mathrm{T}^{-1}]^2}{[\mathrm{T}^2]}$<br><br>$\left[\dfrac{\alpha^2}{\beta}\right] = \dfrac{[\mathrm{M}^2 \mathrm{L}^3 \mathrm{T}^{-2}]}{[\mathrm{T}^2]} = [\mathrm{M}^2 \mathrm{L}^3 \mathrm{T}^{-4}]$<br><br>Answer: \left[\mathrm{M}^2 \mathrm{L}^3 \mathrm{T}^{-4}\right]$