# cube root

The cube root of a real number $x$, written as $\sqrt{x}$, is the real number $y$ such that $y^{3}=x$. Equivalently, $\sqrt{x}^{3}=x$. Or, $\sqrt{x}\sqrt{x}\sqrt{x}=x$. The cube root notation is actually an alternative to exponentiation. That is, $\sqrt{x}=x^{\frac{1}{3}}$.

Examples:

1. 1.

$\sqrt{-8}=-2$ because $(-2)^{3}=(-2)\times(-2)\times(-2)=-8$.

2. 2.

$\sqrt{x^{3}+3x^{2}+3x+1}=x+1$ because $(x+1)^{3}=(x+1)(x+1)(x+1)=(x^{2}+2x+1)(x+1)=x^{3}+3x^{2}+3x+1$.

3. 3.

$\sqrt{x^{3}y^{3}}=xy$ because $(xy)^{3}=xy\times xy\times xy=x^{3}y^{3}$.

4. 4.

$\sqrt{\frac{8}{125}}=\frac{2}{5}$ because $(\frac{2}{5})^{3}=\frac{2^{3}}{5^{3}}=\frac{8}{125}$.

 Title cube root Canonical name CubeRoot Date of creation 2013-03-22 11:57:22 Last modified on 2013-03-22 11:57:22 Owner Daume (40) Last modified by Daume (40) Numerical id 12 Author Daume (40) Entry type Definition Classification msc 11-00 Related topic NthRoot Related topic SquareRoot Related topic RationalNumber Related topic IrrationalNumber Related topic RealNumber Related topic ComplexNumber Related topic CubeOfANumber