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# Square and cube roots

The square root of a number is a value that, when squared, equals the original number.

A positive number has two opposite square roots.

$4$ has two square roots $2$ and $-2$, because $$2^\Tred{2} = \Tblue{4},\quad (-2)^\Tred{2} = \Tblue{4}.$$

Zero has one square root. A negative number has no real square root.

$-2$ has no square root. $0$ is the only square root of $0$!

The positive square root of $a$ is written $\sqrt{a}$.

The square roots of $2$ are $\sqrt{2}$ and $-\sqrt{2}$.

The cube root of a number is the value that, when cubed, equals the original number. The cube root of $a$ is written $\sqrt[3]{a}$.

Positive and negative numbers have exactly one cube root.

$$\sqrt[\Tred{3}]{\Tblue{64}} = 4,\quad 4^\Tred{3} = \Tblue{64},\qquad \sqrt[\Tred{3}]{\Tblue{-27}} = -3,\quad (-3)^{\Tred{3}} = \Tblue{-27}.$$
The side length of a square is the positive square root of its area.