The potential energy U of a system as a function of distance x is...

By the principle of dimensional homogeneity, all terms in a physical equation must have the same dimensions.

Therefore, the dimensions of $\dfrac{A}{x}$ and B x^2 must be equal to the dimensions of potential energy $U$.

\left[\dfrac{A}{x}\right] = [B x^2]$<br><br>$\dfrac{[A]}{[B x^3]} = [M^0 L^0 T^0]$<br><br>Thus, the quantity$\dfrac{A}{B x^3}$is dimensionless.<br><br>Answer:$\dfrac{A}{B x^3}