Algebra

Squaring both sides, we get that \(\left(m + \frac 1m \right)^2 = 8^2\). Expanding gives \(m^2 + \frac{1}{m^2} + 2 = 8^2\), so hence \(m^2 + \frac{1}{m^2} = 8^2 - 2 = 62\). Squaring both sides again, we get that \(\left(m^2 + \frac{1}{m^2}\right)^2 = 62^2\). Expanding gives \(m^4 + \frac{1}{m^4} + 2 = 62^2\), so hence \(m^4 + \frac{1}{m^4} = 62^2 - 2 = \boxed{3842}\).