\[ R=\frac{m_{\mathrm{e}} k^{2} e^{4}}{4 \pi c \hbar^{3}} \] Solve for \( e \)

To solve for \( e \) in the equation \( R = \frac{m_{\mathrm{e}} k^{2} e^{4}}{4 \pi c \hbar^{3}} \), we start by isolating \( e^{4} \): 1. Multiply both sides by \( 4 \pi c \hbar^{3} \): \[ R \cdot 4 \pi c \hbar^{3} = m_{\mathrm{e}} k^{2} e^{4} \] 2. Divide both sides by \( m_{\mathrm{e}} k^{2} \): \[ e^{4} = \frac{R \cdot 4 \pi c \hbar^{3}}{m_{\mathrm{e}} k^{2}} \] 3. Finally, take the fourth root of both sides to solve for \( e \): \[ e = \left( \frac{R \cdot 4 \pi c \hbar^{3}}{m_{\mathrm{e}} k^{2}} \right)^{1/4} \] So, we have the solution for \( e \): \[ e = \left( \frac{R \cdot 4 \pi c \hbar^{3}}{m_{\mathrm{e}} k^{2}} \right)^{1/4} \]