Energy of a complex exponential signal
Just apply Euler’s formula. Hence

$$|e^{at + jbt}|^2 = |e^{at} e^{jbt}|^2 = |e^{at} [\cos(bt) + j \sin(b t)]|^2.$$

Thus

$$|e^{at + jbt}|^2 = e^{2at} [\cos^2(b t) + \sin^2(b t)] = e^{2at}.$$