1. **State the problem:** We have the equation $$V = P^{1/2} + \frac{1}{n} \times \frac{1}{x}$$ where $V$ is volume, $P$ is pressure, and we need to find the dimensions of $n$ and $x$. 2. **Recall dimensions:** - Volume $V$ has dimensions $[L^3]$. - Pressure $P$ has dimensions $[M L^{-1} T^{-2}]$. 3. **Analyze the first term:** $$P^{1/2}$$ Dimensions of $P^{1/2}$ are $$[M L^{-1} T^{-2}]^{1/2} = [M^{1/2} L^{-1/2} T^{-1}]$$ 4. **Since $V$ has dimension $[L^3]$, the sum $P^{1/2} + \frac{1}{n} \times \frac{1}{x}$ implies both terms must have the same dimensions.** 5. **Set dimensions of second term equal to $[L^3]$:** $$\frac{1}{n} \times \frac{1}{x} = [L^3]$$ 6. **Rewrite:** $$\frac{1}{n x} = [L^3] \implies n x = [L^{-3}]$$ 7. **Assume $n$ and $x$ have dimensions $[N]$ and $[X]$ respectively, so:** $$[N][X] = [L^{-3}]$$ 8. **Since $n$ and $x$ are unknown, we need another relation. From the sum, the first term $P^{1/2}$ has dimension $[M^{1/2} L^{-1/2} T^{-1}]$ and the second term $\frac{1}{n x}$ has dimension $[L^3]$. They cannot be added unless they have the same dimension.** 9. **Therefore, the problem implies $V$ is dimensionally consistent only if the terms have the same dimension. So, the problem likely means the terms are added with coefficients or the expression is symbolic.** 10. **If we consider the terms separately, the dimensions of $n$ and $x$ satisfy:** $$n x = [L^{-3}]$$ **Hence, the product of the dimensions of $n$ and $x$ is $[L^{-3}]$. Without additional information, the individual dimensions cannot be uniquely determined.**