1. **State the problem:** We have the equation $$V = P^{1/2} + \frac{1}{n} \times \frac{1}{x}$$ where $V$ is velocity, $P$ is pressure, and we need to find the dimensions of $n$ and $x$. 2. **Recall the dimensions:** - Velocity $V$ has dimensions $$[V] = LT^{-1}$$ - Pressure $P$ has dimensions $$[P] = ML^{-1}T^{-2}$$ 3. **Analyze the first term:** $$P^{1/2}$$ means the square root of pressure, so $$[P^{1/2}] = \left(ML^{-1}T^{-2}\right)^{1/2} = M^{1/2}L^{-1/2}T^{-1}$$ 4. **Since $V$ is velocity, the dimensions of the entire right side must be equal to $[V] = LT^{-1}$.** 5. **Look at the second term:** $$\frac{1}{n} \times \frac{1}{x} = \frac{1}{nx}$$ 6. **The dimensions of the second term must also be $$LT^{-1}$$ to be consistent with $V$. So:** $$\left[\frac{1}{nx}\right] = LT^{-1}$$ 7. **Rewrite this as:** $$[n][x] = \frac{1}{LT^{-1}} = L^{-1}T$$ 8. **From step 3, the first term has dimensions $$M^{1/2}L^{-1/2}T^{-1}$$ which is not the same as velocity. This means the two terms cannot be added unless their dimensions are the same. So, for the equation to be dimensionally consistent, the first term must have dimensions of velocity:** $$M^{1/2}L^{-1/2}T^{-1} = LT^{-1}$$ This is not true unless $M^{1/2}L^{-1/2} = L$, which is impossible. So the problem likely assumes the terms are dimensionally consistent separately. 9. **Therefore, the second term must have dimensions of velocity:** $$\frac{1}{nx} = LT^{-1} \implies [n][x] = L^{-1}T$$ 10. **We cannot find unique dimensions for $n$ and $x$ individually without more information, but their product must have dimensions:** $$[n][x] = L^{-1}T$$ **Final answer:** - Dimensions of $n$ and $x$ satisfy $$[n][x] = L^{-1}T$$ - Velocity $V$ has dimensions $$LT^{-1}$$ - Pressure $P$ has dimensions $$ML^{-1}T^{-2}$$