If the dimensions of length are expressed as ${G}^{x}{c}^{y}{h}^{z}$; where G, c and h are the universal gravitational constant, speed of light and Planck's constant respectively, then

(1) $x=\frac{1}{2},\text{\hspace{0.17em}\hspace{0.17em}}y=\frac{1}{2}$

(2) $x=\frac{1}{2},\text{\hspace{0.17em}\hspace{0.17em}}z=\frac{1}{2}$

(3) $y=\frac{1}{2},\text{\hspace{0.17em}\hspace{0.17em}}z=\frac{3}{2}$

(4) $y=-\frac{3}{2},\text{\hspace{0.17em}\hspace{0.17em}}z=\frac{1}{2}$

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#1 | Basic Concepts & Examples
#2 | Dimensional Analysis : Remaining

Concept Questions :-

Dimensions

(2, 4) Length $\propto$ Gxcyhz

L= ${\left[{M}^{-1}{L}^{3}{T}^{-2}\right]}^{x}\text{\hspace{0.17em}}$${\left[L{T}^{-1}\right]}^{y}{\left[M{L}^{2}{T}^{-1}\right]}^{z}$

By comparing the power of M, L and T in both sides we get $-x+z=0$, $3x+y+2z=1$ and $-2x-y-z=0$

By solving above three equations we get

$x=\frac{1}{2},\text{\hspace{0.17em}}y=-\frac{3}{2},z=\frac{1}{2}$

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