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 :

Options

A. `x=1/2,y=1/2`
B. `x=1/2, z=1/2`
C. `y=1/2,z=3/2`
D. `y=-3/2, z=1/2`

Answer

To solve the problem, we need to express the dimensions of length in terms of the universal gravitational constant (G), the speed of light (c), and Planck's constant (h). We will derive the dimensions for each of these constants and then set up equations based on the dimensional analysis.

### Step-by-Step Solution:

1. **Identify the dimensions of G, c, and h:**
- The universal gravitational constant \( G \) is defined from Newton's law of gravitation:
\[ F = \frac{G m_1 m_2}{r^2} \]
Rearranging gives us:
\[ G = \frac{F r^2}{m_1 m_2} \]
The dimensions of force \( F \) are \( [M L T^{-2}] \), and the dimensions of distance \( r \) are \( [L] \). Thus, the dimensions of \( G \) are:
\[ [G] = \frac{[M L T^{-2}] [L^2]}{[M^2]} = [M^{-1} L^3 T^{-2}] \]

- The speed of light \( c \) has dimensions:
\[ [c] = [L T^{-1}] \]

- Planck's constant \( h \) relates energy and frequency:
\[ E = h \nu \implies h = \frac{E}{\nu} \]
The dimensions of energy \( E \) are \( [M L^2 T^{-2}] \) and frequency \( \nu \) has dimensions \( [T^{-1}] \). Thus, the dimensions of \( h \) are:
\[ [h] = \frac{[M L^2 T^{-2}]}{[T^{-1}]} = [M L^2 T^{-1}] \]

2. **Express the dimensions of length \( L \) in terms of \( G, c, h \):**
We express the dimensions of length \( L \) as:
\[ L = G^x c^y h^z \]

Substituting the dimensions we found:
\[ [L] = [G]^x [c]^y [h]^z = ([M^{-1} L^3 T^{-2}])^x ([L T^{-1}])^y ([M L^2 T^{-1}])^z \]

This expands to:
\[ [L] = [M^{-x} L^{3x} T^{-2x}] [L^y T^{-y}] [M^z L^{2z} T^{-z}] \]

Combining the dimensions:
\[ [L] = [M^{-x + z} L^{3x + y + 2z} T^{-2x - y - z}] \]

3. **Set up equations based on dimensional analysis:**
For the dimensions to match \( [L] \) (which is \( [L^1] \)), we need:
- Coefficient of \( M \): \( -x + z = 0 \) (1)
- Coefficient of \( L \): \( 3x + y + 2z = 1 \) (2)
- Coefficient of \( T \): \( -2x - y - z = 0 \) (3)

4. **Solve the equations:**
From equation (1):
\[ z = x \]

Substitute \( z = x \) into equations (2) and (3):
- From (2):
\[ 3x + y + 2x = 1 \implies 5x + y = 1 \quad \text{(4)} \]
- From (3):
\[ -2x - y - x = 0 \implies -3x - y = 0 \implies y = -3x \quad \text{(5)} \]

Substitute (5) into (4):
\[ 5x - 3x = 1 \implies 2x = 1 \implies x = \frac{1}{2} \]

Then, using \( x = \frac{1}{2} \) in (1):
\[ z = \frac{1}{2} \]

And using \( x = \frac{1}{2} \) in (5):
\[ y = -3 \left(\frac{1}{2}\right) = -\frac{3}{2} \]

5. **Final values:**
- \( x = \frac{1}{2} \)
- \( y = -\frac{3}{2} \)
- \( z = \frac{1}{2} \)

### Conclusion:
The dimensions of length expressed as \( G^x c^y h^z \) yield:
- \( x = \frac{1}{2}, y = -\frac{3}{2}, z = \frac{1}{2} \)