Cube Root
ArithmeticThe cube root of a number is the value that, when multiplied by itself three times, gives that number.
Formula
\sqrt[3]{n} = x \text{ means } x^3 = n
Definition
The cube root of a number is what you multiply by itself three times to get that number. We write it with a 3 inside the radical sign.
Example
The cube root of $8$ is $2$, because $2 \times 2 \times 2 = 8$. The cube root of $27$ is $3$, because $3 \times 3 \times 3 = 27$.
Key Insight
The cube root is the side length of a cube that has the given volume. A cube with volume $64$ cubic units has side length $4$ (because $4^3 = 64$).
Definition
The cube root of $n$ is the real number $x$ such that $x^3 = n$, written $\sqrt[3]{n}$ or $n^{1/3}$. Unlike square roots, cube roots exist for negative numbers: $\sqrt[3]{-8} = -2$ because $(-2)^3 = -8$. Every real number has exactly one real cube root.
Example
$\sqrt[3]{125} = 5$. $\sqrt[3]{-27} = -3$. $\sqrt[3]{10} \approx 2.154$ (irrational). Simplify $\sqrt[3]{54}$: $54 = 27 \times 2 = 3^3 \times 2$. So $\sqrt[3]{54} = 3\sqrt[3]{2}$.
Key Insight
The cube root is an odd function: $\sqrt[3]{-x} = -\sqrt[3]{x}$. This is because $(-x)^3 = -(x^3)$, so the cube root of a negative is the negative of the cube root of the positive.
Definition
The real cube root $x^{1/3}$ for real $x$ is well-defined and continuously differentiable for $x \neq 0$: $\frac{d}{dx}(x^{1/3}) = \frac{1}{3}x^{-2/3}$. In the complex plane, every non-zero $z$ has three cube roots: $|z|^{1/3} e^{i(\arg(z)+2\pi k)/3}$ for $k = 0, 1, 2$. These three roots form the vertices of an equilateral triangle in the complex plane.
Example
Cube roots of $1$ (unity): $1$, $e^{2\pi i/3} = -1/2 + i\sqrt{3}/2$, $e^{4\pi i/3} = -1/2 - i\sqrt{3}/2$. Their sum is $0$, and their product is $1$.
Key Insight
The $n$-th roots of unity are evenly spaced on the unit circle in the complex plane. This geometric picture underlies the discrete Fourier transform and explains why DFT algorithms (FFT) are structured as they are.