= Solution
Convert both inputs to <energy density>. The matter contribution, including rest energy, is $u_{M,0}=\rho_{M,0}c^2=2.25\times10^{-10}\,\mathrm{J\,m^{-3}}$. The given photon value is $u_{\gamma,0}=5\times10^5(1.602\times10^{-19})=8.01\times10^{-14}\,\mathrm{J\,m^{-3}}$. Hence
$$
\boxed{\frac{u_{M,0}}{u_{\gamma,0}}\simeq2.80\times10^3.}
$$
Matter <energy density> scales as $(1+z)^3$, whereas the <cosmic microwave background> scales as $(1+z)^4$. The <photon-only matter-radiation equality> condition is therefore
$$
1+z_{\rm eq}=\frac{u_{M,0}}{u_{\gamma,0}},\qquad
\boxed{z_{\rm eq}\simeq2.8\times10^3.}
$$
This uses the supplied CMB photon density. Additional relativistic species change equality for total radiation.
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