In AC circuits, resistance and reactances have the same unit.

• Resistance: $R$ has unit ohm $(\Omega)$
• Inductive reactance: $X_L = \omega L$ also has unit ohm $(\Omega)$
• Capacitive reactance: $X_C = \dfrac{1}{\omega C}$ also has unit ohm $(\Omega)$

Hence, $[R] = [X_L] = [X_C] = \Omega$

Now check each option dimensionally.

Option A: $\frac{R}{X_L X_C}$ Dimensions: $\frac{\Omega}{\Omega\cdot\Omega} = \Omega^{-1}$ Not dimensionless.

Option B: $R X_L X_C$ Dimensions: $\Omega\cdot\Omega\cdot\Omega = \Omega^3$ Not dimensionless.

Option C: $\frac{R}{\sqrt{X_L X_C}}$ Dimensions: $\frac{\Omega}{\sqrt{\Omega\cdot\Omega}} = \frac{\Omega}{\Omega} = 1$ So this is dimensionless.

Option D: $R\frac{X_L}{X_C}$ Dimensions: $\Omega\cdot \frac{\Omega}{\Omega} = \Omega$ Not dimensionless.

Therefore, the only dimensionless quantity is $\boxed{\frac{R}{\sqrt{X_L X_C}}}$

Comparison with stored correct answer:

• Derived answer: C
• Stored correct answer: C

They match.