1. Write dimensions of all quantities

We are given $m = k\, c^P G^{-1/2} h^{1/2}$ where $k$ is dimensionless.

So dimensions must satisfy $[M] = [c]^P [G]^{-1/2} [h]^{1/2}$

Now,

• Speed of light: $[c] = LT^{-1}$
• Gravitational constant: $[G] = M^{-1}L^3T^{-2}$
• Planck's constant: $[h] = ML^2T^{-1}$

2. Find dimensions of each factor

$[c]^P = L^P T^{-P}$

$[G]^{-1/2} = \left(M^{-1}L^3T^{-2}\right)^{-1/2} = M^{1/2}L^{-3/2}T^1$

$[h]^{1/2} = \left(ML^2T^{-1}\right)^{1/2} = M^{1/2}L^1T^{-1/2}$

3. Multiply all dimensions

$[c]^P [G]^{-1/2} [h]^{1/2} = \left(L^P T^{-P}\right)\left(M^{1/2}L^{-3/2}T^1\right)\left(M^{1/2}L^1T^{-1/2}\right)$

Combine powers:

• Mass: $M^{1/2+1/2} = M^1$

• Length: $L^{P-3/2+1} = L^{P-1/2}$

• Time: $T^{-P+1-1/2} = T^{-P+1/2}$

Thus RHS has dimensions $M^1 L^{P-1/2} T^{-P+1/2}$

4. Compare with dimensions of mass

Since LHS is mass, $[M] = M^1L^0T^0$

So, $P - \frac{1}{2} = 0$ $-P + \frac{1}{2} = 0$

Both give $P = \frac{1}{2}$

5. Check options

The correct option is:

C: $\frac{1}{2}$