If the velocity of light $$c$$, universal gravitational constant $$G$$ and planck's constant $$h$$ are chosen as fundamental quantities. The dimensions of mass in the new system is:

We need to express mass $$M$$ in terms of $$c$$, $$G$$, and $$h$$.

Dimensions: $$[c] = LT^{-1}$$, $$[G] = M^{-1}L^3T^{-2}$$, $$[h] = ML^2T^{-1}$$

Let $$[M] = [h]^a [c]^b [G]^d$$. Comparing dimensions:

Mass: $$a - d = 1$$ ... (1)

Length: $$2a + b + 3d = 0$$ ... (2)

Time: $$-a - b - 2d = 0$$ ... (3)

From (1): $$a = 1 + d$$

From (3): $$b = -a - 2d = -(1+d) - 2d = -1 - 3d$$

Substituting in (2): $$2(1+d) + (-1-3d) + 3d = 0 \Rightarrow 1 + 2d = 0 \Rightarrow d = -\frac{1}{2}$$

Therefore: $$a = \frac{1}{2}$$, $$b = \frac{1}{2}$$, $$d = -\frac{1}{2}$$