Root Calculator

Understanding Roots and Radicals

A radical root expression represents the inverse operation of exponentiation. The symbol $\sqrt{\phantom{x}}$ denotes a radical, the number inside the radical is the **radicand** ($x$), and the small number outside is the **degree** or **index** ($n$).

Real vs. Complex Roots

Taking the root of a negative radicand depends entirely on whether the degree of the root is odd or even:

• Odd Roots (e.g. 3rd, 5th, 7th root): Real numbers always have a single real root. For example, $\sqrt[3]-8 = -2$ since $(-2) \times (-2) \times (-2) = -8$.
• Even Roots (e.g. Square Root, 4th root): Real negative numbers do not have real roots, because squaring any real number yields a positive result. Instead, they resolve to complex numbers. For example, $\sqrt-4 = 2i$, where $i = \sqrt-1$ is the imaginary unit.

Finding Complex Roots with Euler's Formula

For even root degrees of negative values, the principal root is calculated on the complex plane using Euler's formula: $$r(\cos\theta + i\sin\theta)$$ Where the magnitude $r = |x|^{1/n}$ and the angle $\theta = \frac{\pi}{n}$. This represents the primary rotation on the complex coordinate grid.