1. **Problem statement:** Given the velocity equation $$v = \sqrt{P + \frac{1}{x^n}}$$ where $P$ is pressure, find the dimensions of $n$ and $x$. 2. **Recall dimensions:** Velocity $v$ has dimensions $[L T^{-1}]$, pressure $P$ has dimensions $[M L^{-1} T^{-2}]$, and $x$ is a variable with unknown dimension $[X]$. The exponent $n$ is dimensionless because exponents must be pure numbers. 3. **Analyze the equation inside the square root:** The expression inside the square root must have the same dimension because you cannot add quantities of different dimensions. So, $$[P] = \left[\frac{1}{x^n}\right]$$ 4. **Express the dimension of $\frac{1}{x^n}$:** $$\left[\frac{1}{x^n}\right] = [X]^{-n}$$ 5. **Equate dimensions inside the root:** $$[P] = [X]^{-n}$$ $$[M L^{-1} T^{-2}] = [X]^{-n}$$ 6. **Dimension of velocity $v$ is:** $$[v] = [L T^{-1}] = \sqrt{[P]} = \sqrt{[M L^{-1} T^{-2}]} = [M^{1/2} L^{-1/2} T^{-1}]$$ 7. **Since $v$ is velocity, the dimension inside the root must be $[v]^2$:** $$[P + 1/x^n] = [v]^2 = [L^2 T^{-2}]$$ 8. **But from step 5, $[P] = [M L^{-1} T^{-2}]$, which is not equal to $[L^2 T^{-2}]$. This is a contradiction unless $P$ is dimensionless or the problem assumes $P$ has dimension $[L^2 T^{-2}]$.** 9. **Assuming $P$ has dimension $[L^2 T^{-2}]$ (like energy per mass), then:** $$[P] = [L^2 T^{-2}] = [X]^{-n}$$ 10. **Therefore:** $$[X]^{-n} = [L^2 T^{-2}]$$ 11. **Taking logarithms of dimensions, we get:** $$-n \cdot \log [X] = \log [L^2 T^{-2}]$$ 12. **Since $n$ is dimensionless, $[X]$ must have dimensions such that:** $$[X]^n = [L^{-2} T^{2}]$$ 13. **If we choose $[X] = [L^a T^b]$, then:** $$[X]^n = [L^{a n} T^{b n}] = [L^{-2} T^{2}]$$ 14. **Equate exponents:** $$a n = -2$$ $$b n = 2$$ 15. **Since $n$ is dimensionless, $a$ and $b$ can be any real numbers satisfying:** $$a = \frac{-2}{n}, \quad b = \frac{2}{n}$$ 16. **Summary:** - $n$ is dimensionless. - $x$ has dimension $[L^{a} T^{b}]$ with $a = -2/n$ and $b = 2/n$. **Final answer:** $$\boxed{n \text{ is dimensionless}, \quad [x] = [L^{-2/n} T^{2/n}]}$$