Exponentiation

Common term for three tightly linked concepts:

Powers

$base^{exponent} = power$

Roots

$\sqrt[exponent]{power} = base$

Logarithms

$log_{base}(power) = exponent$

In General Form

$x^{y} = z \Leftrightarrow \sqrt[y]{z} = x \Leftrightarrow log_x(z) = y$

Example

$x = 10$

$y = 3$

$z = 1000$

$10^3 = 1000 \Leftrightarrow \sqrt[3]{1000} = 10 \Leftrightarrow log_{10}(1000) = 3$

Inverse Exponents?

$x^{-y} = \frac{1}{x^y}, x \neq 0$

Hating Those Roots?

$\sqrt[y]{z^x} = z^{\frac{x}{y}}$

Complex Numbers

$\abs{z} = \abs{zhat}$

$\abs{z} = \sqrt{a^2 + (-b)^2} = \sqrt{a^2 + b^2} = \abs{zhat}$

$z * z^{-1} = 1$

$(a + ib)(u + iv) = aU - bv + i(av + ub) = 1 = 1 + 0i$

$au - bv = 1$

$au + ub = 0$

$au -b(-\frac{ub}{a}) = 1$

$v = -\frac{ub}{a}$

$u(a + \frac{b^2}{a}) = 1$

$u(\frac{a^2 + b^2}{a}) = 1$

$u = \frac{a}{a^2 + b^2}$

$v = -\frac{b}{a^2+b^2}$