Dimensional consistency means that each term of an equation must have the same dimensions. Therefore any terms expressing an area should have the dimension of $L^2$:

$2pi r =L Rightarrow$ This does not conform to an Area.

$4pi r^2=L^2 Rightarrow$ This is the dimension of an Area.

The surface area of a sphere is $4pi r^2$

$x$ Distance $Rightarrow$ Dimension: $L$

$t$ Time$Rightarrow$ Dimension: $T$

$v$ Speed $Rightarrow$ Dimension $frac{L}{T}$

$a$ Acceleration $Rightarrow$ Dimension $frac{L}{T^2}$

(a) $frac{x}{t^2}Rightarrow frac{L}{T^2} Rightarrow$ Acceleration.

(b) $vtRightarrow frac{L}{T}T=L Rightarrow$ Distance.

(c) $(2ax)^{frac{1}{2}}Rightarrow (frac{L}{T^2}L)^{frac{1}{2}}=(frac{L^2}{T^2})^{frac{1}{2}}=frac{L}{T} Rightarrow$ Speed.

(d)$frac{1}{2}at^2 Rightarrow frac{L}{T^2}T^2=L Rightarrow$Distance

for the given sphere radius 0.735 m:

$$
v=frac{4}{3}pi(0.735)^3=1.66text{ m}^3
$$

$$
begin{align*}
x&=frac{1}{2}at^p \
L&=frac{L}{T^2}T^p \
L&=L*T^{P-2}
end{align*}
$$

In order for an equation to be dimenstionnaly consistant, all terms of the equation need to have the same dimesnsions ,this implies the following:

$$
Rightarrow T^{P-2}=T^{0}
$$

$$
Rightarrow P-2=0
$$

$$
P=boxed{2}
$$

We first convert the quantities in question to the desired units $frac{m}{s}$:

$$
3mm=3mmfrac{1m}{10^3mm}=3*10^{-3}m
$$

$$
1month=1monthfrac{1s}{frac{1}{2592000}1month}=2592000s
$$

Then we determine the order of magnitude of each quantity:

$$
3*10^{-3}mRightarrow 10^{-3}
$$

$$
2592000sRightarrow 10^{6}
$$

We can now determine the magnitude of nails growth:

$$
v=frac{d}{t}=frac{10^{-3}}{10^{6}}=10^{-9}frac{m}{s}
$$

$$
1nm=10^{-9}m
$$