= Solution
Use <stellar homology>: dimensionless profiles and composition are fixed, so proportionality constants do not change along the model sequence. Scaling mass conservation and the <stellar hydrostatic equation> gives
$$
\rho_*\propto\frac M{R^3},\qquad P_*\propto\frac{GM^2}{R^4},\qquad
T_*\propto\frac{P_*}{\rho_*}\propto\frac MR,
$$
where the last relation uses the <ideal gas> equation of state with fixed <mean molecular weight>. The specific <stellar energy-generation rate> is $\epsilon\propto\rho T^\eta$, so
$$
L_{\rm nuc}\sim M\epsilon_*\propto M^{\eta+2}R^{-(\eta+3)}.
$$
By <stellar radiative diffusion>, $T_*/R\sim\kappa_*\rho_*L/R^2T_*^3$, hence
$$
L_{\rm rad}\propto\frac{RT_*^4}{\kappa_*\rho_*}
\propto M^{3-\lambda-\nu}R^{3\lambda+\nu}.
$$
Equate the generated and transported <luminosity> in thermal equilibrium. The <homology relations for radiative stars> give
$$
R^{3+\eta+\nu+3\lambda}\propto M^{\eta+\nu+\lambda-1},\qquad
\boxed{R\propto M^X,\quad
X=\frac{\eta+\nu+\lambda-1}{3+\eta+\nu+3\lambda}.}
$$
This division presumes a nonzero denominator. If it vanishes, the scaling balance instead imposes a condition on $M$ or leaves the radius scaling undetermined, depending on the numerator. For later use, the corresponding <mass-luminosity relation> exponent is
$$
Y=\eta+2-(\eta+3)X
=3-\lambda-\nu+(3\lambda+\nu)X.
$$
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