The dimension of power is $\left[{\mathrm{M}}^1{\mathrm{L}}^2{\mathrm{T}}^{\mathrm{x}}\right]$. The value of $\mathrm{x}$ is .

A particle of mass $m$ is located in a region where its potential energy $\left[U\left(x\right)\right]$ depends on the position $x$ as, $U\left(x\right)=\frac{a}{x^{\mathit{2}}}-\frac{b}{x}$. Here, $a$ and $b$ are positive constants. $\left(\mathrm{i}\right)$ Write the dimensional formulae for $a$ and $b$.

$\left(\mathrm{ii}\right)$ If the time period of oscillation, which is calculated from the above formula, is stated by a student as $T=4\mathrm{\pi }a\sqrt{\frac{ma}{b^2}}$, check whether his answer is dimensionally correct.

Two quantities, $A$ and $B$ have different dimensions. Which mathematical operation, given below, is physically meaningful?

A force $F$ is given by, $F\mathit{=}at\mathit{+}b{t}^{\mathit{2}}$, where $t$ is time. What are the dimensions of $a$ and $b$?