G = 6.67 x 10⁻¹¹ Nm²kg⁻² = ........cm³s⁻²g⁻¹

The correct option is 6.67x10⁻⁸cm³s⁻²g⁻¹.

To convert the value of the gravitational constant $G$ from SI units ($\mathrm{N}{\mathrm{m}}^2{\mathrm{kg}}^{-2}$) to CGS units (${\mathrm{cm}}^3{\mathrm{s}}^{-2}{\mathrm{g}}^{-1}$), we can analyze the fundamental dimensions of the units involved.

First, let us express Newton ($\mathrm{N}$), the SI unit of force, in terms of base SI units:
$1\mathrm{N}=1\mathrm{kg}\cdot \mathrm{m}\cdot {\mathrm{s}}^{-2}$

Substitute this back into the SI unit of $G$:
$\mathrm{N}��{\mathrm{m}}^2\cdot {\mathrm{kg}}^{-2}=(\mathrm{kg}\cdot \mathrm{m}\cdot {\mathrm{s}}^{-2})\cdot {\mathrm{m}}^2\cdot {\mathrm{kg}}^{-2}$
Simplifying the powers, we get:
${\mathrm{kg}}^{1-2}\cdot {\mathrm{m}}^{1+2}\cdot {\mathrm{s}}^{-2}={\mathrm{kg}}^{-1}\cdot {\mathrm{m}}^3\cdot {\mathrm{s}}^{-2}$

Now, we convert each of these base SI units to their corresponding CGS units:
1. $1\mathrm{kg}={10}^3\mathrm{g}\Rightarrow 1{\mathrm{kg}}^{-1}=({10}^3\mathrm{g}{)}^{-1}={10}^{-3}{\mathrm{g}}^{-1}$
2.
3. The unit of time remains seconds ($\mathrm{s}$) in both systems: $1{\mathrm{s}}^{-2}=1{\mathrm{s}}^{-2}$

Now, substitute these conversions into the unit expression for $G$:
$1{\mathrm{kg}}^{-1}{\mathrm{m}}^3{\mathrm{s}}^{-2}=({10}^{-3}{\mathrm{g}}^{-1})\cdot ({10}^6{\mathrm{cm}}^3)\cdot (1{\mathrm{s}}^{-2})$
$1{\mathrm{kg}}^{-1}{\mathrm{m}}^3{\mathrm{s}}^{-2}={10}^{-3+6}{\mathrm{cm}}^3{\mathrm{s}}^{-2}{\mathrm{g}}^{-1}={10}^3{\mathrm{cm}}^3{\mathrm{s}}^{-2}{\mathrm{g}}^{-1}$

Now, apply this conversion factor to the given value of $G$:
$G=6.67\times {10}^{-11}{\mathrm{kg}}^{-1}{\mathrm{m}}^3{\mathrm{s}}^{-2}$
$G=6.67\times {10}^{-11}\times {10}^3{\mathrm{cm}}^3{\mathrm{s}}^{-2}{\mathrm{g}}^{-1}$
$G=6.67\times {10}^{-11+3}{\mathrm{cm}}^3{\mathrm{s}}^{-2}{\mathrm{g}}^{-1}$
$G=6.67\times {10}^{-8}{\mathrm{cm}}^3{\mathrm{s}}^{-2}{\mathrm{g}}^{-1}$