The Rydberg constant R for hydrogen is
$R = - [\frac{1}{4 {πε}_{0}}] . \frac{2 π^{2} {me}^{2}}{{Ch}^{2}}$
$R = (\frac{1}{4 {πε}_{0}}) . \frac{2 π^{2} {me}^{4}}{{Ch}^{2}}$
$R = {(\frac{1}{4 {πε}_{0}})}^{2} . \frac{2 π^{2} {me}^{4}}{C^{2} h^{2}}$
$R = {[\frac{1}{4 {πε}_{0}}]}^{2} . \frac{2 π^{2} {me}^{4}}{{Ch}^{3}}$

because R = $\frac{2 π^{2} k^{2} {me}^{4}}{{ch}^{3}} = {(\frac{1}{4 π ε_{0}})}^{2} \frac{2 π^{2} m e^{4}}{c h^{3}}$

Solution

R = {[\frac{1}{4 {πε}_{0}}]}^{2} . \frac{2 π^{2} {me}^{4}}{{Ch}^{3}}

because R =

\frac{2 π^{2} k^{2} {me}^{4}}{{ch}^{3}} = {(\frac{1}{4 π ε_{0}})}^{2} \frac{2 π^{2} m e^{4}}{c h^{3}}

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