= Solution
With the supplied fixed <band gap> and <effective masses>, the <intrinsic carrier concentration> is
$$
n_i(T)=2\left(\frac{k_BT}{2\pi\hbar^2}\right)^{3/2}(m_e^*m_h^*)^{3/4}
e^{-E_g/(2k_BT)}.
$$
Numerically,
$$
n_i(750\,\mathrm K)=8.53\times10^{22}\,\mathrm{m^{-3}},\qquad
n_i(300\,\mathrm K)=1.01\times10^{19}\,\mathrm{m^{-3}}.
$$
For uncompensated, approximately ionized <donor impurities>, <charge neutrality> and the <semiconductor mass-action law> require
$$
n-p=N_D,\qquad np=n_i^2,\qquad
n=\frac{N_D+\sqrt{N_D^2+4n_i^2}}2.
$$
There is no abrupt extrinsic-to-intrinsic transition. An order-of-magnitude lower crossover density is therefore
$$
\boxed{N_D\gtrsim n_i(750\,\mathrm K)\simeq8.5\times10^{22}\,\mathrm{m^{-3}}
=8.5\times10^{16}\,\mathrm{cm^{-3}}.}
$$
Robust extrinsic dominance requires $N_D\gg n_i$; at the nominal equality, $n=1.618N_D$ and $p=0.618N_D$, so intrinsic contributions are already appreciable. For a quantitative tolerance $p/n\leq\eta$, the minimum ionized <dopant density> is $n_i(1-\eta)/\sqrt\eta$. For example, $\eta=0.1$ would require $N_D\geq2.85n_i$ at the upper temperature.
Using the nominal crossover estimate and the complete-ionization approximation at room temperature, $n\simeq N_D$ and
$$
\boxed{p(300\,\mathrm K)\simeq\frac{n_i(300\,\mathrm K)^2}{N_D}
=1.19\times10^{15}\,\mathrm{m^{-3}}\simeq1.19\times10^9\,\mathrm{cm^{-3}}.}
$$
For the twofold neutral-donor degeneracy, a consistency calculation gives about $91.3\%$ ionization at $300\,\mathrm K$, raising this minority density to $1.30\times10^{15}\,\mathrm{m^{-3}}$, about $9.6\%$ above the complete-ionization estimate. Stronger extrinsic dominance uses more donors and correspondingly fewer <electron holes>. Actual germanium has <temperature>-dependent band parameters and multiple valleys; the numerical estimate consistently uses the fixed effective data supplied rather than adding an unsupplied multiplicity.
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