1. **Stating the problem:** We have the equation $$V = P^{1/2} + \frac{1}{n} \times x$$ where $V$ is volume, $P$ is pressure, and we need to find the dimensions of $n$ and $x$. 2. **Recall dimensional formulas:** - Volume $V$ has dimension $[L^3]$. - Pressure $P$ has dimension $[M L^{-1} T^{-2}]$. 3. **Analyze the equation:** Since $V$ and $P^{1/2} + \frac{1}{n} x$ are added, both terms must have the same dimension as $V$. 4. **Dimension of $P^{1/2}$:** $$[P^{1/2}] = \left([M L^{-1} T^{-2}]\right)^{1/2} = [M^{1/2} L^{-1/2} T^{-1}]$$ 5. **Equate dimensions of $P^{1/2}$ and $V$:** They must be equal for addition, so $$[M^{1/2} L^{-1/2} T^{-1}] = [L^3]$$ This is not possible unless the terms are dimensionally consistent separately, so the equation implies the terms are dimensionally independent and the sum is symbolic or the problem expects $\frac{1}{n} x$ to have dimension $[L^3]$. 6. **Dimension of $\frac{1}{n} x$:** Since $V$ has dimension $[L^3]$, $$\left[\frac{1}{n} x\right] = [L^3]$$ 7. **Assuming $x$ has dimension $[X]$ and $n$ has dimension $[N]$, then:** $$\frac{[X]}{[N]} = [L^3] \implies [X] = [N] [L^3]$$ 8. **To find $[N]$ and $[X]$, we use the fact that $P^{1/2}$ and $V$ are added, so the problem likely means $P^{1/2}$ and $\frac{1}{n} x$ have the same dimension:** $$[P^{1/2}] = \left[\frac{1}{n} x\right] = [L^3]$$ 9. **From step 4, $[P^{1/2}] = [M^{1/2} L^{-1/2} T^{-1}]$, so:** $$[L^3] = [M^{1/2} L^{-1/2} T^{-1}]$$ This is a contradiction, so the problem likely means $V$ equals the sum of two terms with different dimensions, which is physically inconsistent unless $V$ is not volume or the problem is symbolic. 10. **Alternatively, if $V$ has dimension $[M^{1/2} L^{-1/2} T^{-1}]$ (same as $P^{1/2}$), then:** $$[V] = [P^{1/2}] = [M^{1/2} L^{-1/2} T^{-1}]$$ 11. **Then, since $V = P^{1/2} + \frac{1}{n} x$, both terms must have dimension $[M^{1/2} L^{-1/2} T^{-1}]$, so:** $$\left[\frac{1}{n} x\right] = [M^{1/2} L^{-1/2} T^{-1}]$$ 12. **Let $[x] = [X]$ and $[n] = [N]$, then:** $$\frac{[X]}{[N]} = [M^{1/2} L^{-1/2} T^{-1}] \implies [X] = [N] [M^{1/2} L^{-1/2} T^{-1}]$$ 13. **Since $n$ is a constant or parameter, it can be dimensionless or have dimension to make $x$ have a desired dimension. Without more info, the dimensions are:** - $[n] = [N]$ (unknown) - $[x] = [N] [M^{1/2} L^{-1/2} T^{-1}]$ **Final answer:** - Dimensions of $n$ are arbitrary $[N]$. - Dimensions of $x$ are $[N] [M^{1/2} L^{-1/2} T^{-1}]$. This means $n$ and $x$ are related by the above dimensional formula to keep the equation dimensionally consistent.