State the principle of homogeneity of dimensions.

If energy is given by $U = \frac{A\sqrt{x}}{x^2 + B}$,then the dimensions of $AB$ are:

Planck's constant $(h)$,speed of light in vacuum $(c)$,and Newton's gravitational constant $(G)$ are three fundamental constants. Which of the following combinations of these has the dimension of length?

The $SI$ unit of energy is $J = kg \, m^{2} \, s^{-2}$; that of speed $v$ is $m \, s^{-1}$ and of acceleration $a$ is $m \, s^{-2}$. Which of the formulae for kinetic energy $(K)$ given below can you rule out on the basis of dimensional arguments ($m$ stands for the mass of the body):
$(a)$ $K = m^{2} v^{3}$
$(b)$ $K = (1/2) m v^{2}$
$(c)$ $K = m a$
$(d)$ $K = (3/16) m v^{2}$
$(e)$ $K = (1/2) m v^{2} + m a$

If the velocity of light $c$,universal gravitational constant $G$,and Planck's constant $h$ are chosen as fundamental quantities,the dimensions of mass in the new system are:

An experiment measures quantities $a, b,$ and $c$ and then $x$ is calculated from $x = \frac{a^{1/2} b^2}{c^3}.$ If the percentage errors in $a, b,$ and $c$ are $\pm 1\%, \pm 3\%,$ and $\pm 2\%$ respectively,then the percentage error in $x$ is: