How do you rearrange
$$\frac{m_{1}}{\gamma_{2}} = \frac{\sqrt{\frac{m_1}{\gamma_2}}-m_1}{\gamma_1}$$
to get
$$\frac{\gamma_1}{\gamma_2} = \sqrt\frac{1}{m_1\gamma_2} - 1 $$ ?
I understand this may be remedial but if someone here could show me the steps then that would be very helpful.
Multiplying on both sides by $\gamma_1$ and dividing by $m_1$ yields the desired result. $$\frac{m_{1}}{\gamma_{2}} = \frac{\sqrt{\frac{m_1}{\gamma_2}}-m_1}{\gamma_1} \Leftrightarrow \frac{\gamma_1}{\gamma_2}=\frac{1}{m_1}\left(\sqrt{\frac{m_1}{\gamma_2}}-m_1\right) \Leftrightarrow \frac{\gamma_1}{\gamma_2} = \sqrt\frac{1}{m_1\gamma_2} - 1 $$
Multiply both sides by $\gamma_1$ and divide by $m_1$ to get $$\frac{\gamma_1}{\gamma_2} = \frac{1}{m_1}\left(\sqrt{\frac{m_1}{\gamma_2}} - m_1\right)$$ Now distribute the multiplication, remembering that $m_1 = \sqrt{m_1^2}$ (provided that $m_1$ is positive) and you get the result.