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Step-by-Step Solution
Step 1: Identify the Required Quantity
We are asked to form a quantity with the dimension of length using the three fundamental constants: Planck's constant $h$, speed of light $c$, and Newton's gravitational constant $G$.
Step 2: Assign General Exponents to Each Constant
Let the required length $L$ be proportional to:
$$
L \propto h^{p} \, c^{q} \, G^{r}.
$$
We wish to determine the values of $p$, $q$, and $r$ so that the resulting combination has the dimension of length.
Step 3: Write Down the Dimensional Forms of Each Constant
The dimensions of each constant are as follows:
Planck's constant $h$:
$$
[h] = M L^2 T^{-1}
$$
Speed of light $c$:
$$
[c] = L T^{-1}
$$
Newton's gravitational constant $G$:
$$
[G] = M^{-1} L^{3} T^{-2}
$$
Step 4: Express the Overall Dimensions
The overall dimensions of $L \propto h^{p} c^{q} G^{r}$ become:
$$
\bigl[M L^2 T^{-1}\bigr]^{p} \, \bigl[L T^{-1}\bigr]^{q} \, \bigl[M^{-1} L^{3} T^{-2}\bigr]^{r}.
$$
When multiplied out, this expression should yield the dimension of length, which is:
$$
[M^0 L^1 T^0].
$$
Step 5: Compare Exponents of M, L, and T
To match the dimension $[M^0 L^1 T^0]$, we set up equations by equating the powers of $M$, $L$, and $T$ on both sides:
Power of M:
$$
p + 0 + (-1)r = 0
\quad \Rightarrow \quad p - r = 0
\quad (1)
$$
Power of L:
$$
2p + q + 3r = 1
\quad (2)
$$
Power of T:
$$
- p + (-1)q + (-2)r = 0
\quad \Rightarrow \quad -p - q - 2r = 0
\quad (3)
$$
Step 6: Solve the System of Equations
We solve equations (1), (2), and (3) simultaneously:
From (1): $p - r = 0 \implies p = r$.
Substitute $p = r$ into (2) and (3) to find $p, q, r$.
After solving, we get:
$$
p = r = \frac{1}{2}, \quad q = -\frac{3}{2}.
$$
Step 7: Write the Final Expression
Substituting $p = \tfrac{1}{2}$, $q = -\tfrac{3}{2}$, and $r = \tfrac{1}{2}$ back, we obtain the combination that has the dimension of length:
$$
L = \frac{\sqrt{h\,G}}{c^{3/2}}.
$$
This matches the provided correct option
$$
\frac{\sqrt{hG}}{c^{3/2}}.
$$
