Square root

A square root of a number $x$ is a number $y$ such that $y^2 = x$ . Thus $y$ is a square root of $x$ if and only if $x$ is the square of $y$ .

The square root (or the principle square root) of a number $x$ is denoted $\sqrt x$ . For instance, $\sqrt 4 = 2$ . When we consider only positive reals, the square root function is the inverse of the squaring function. However, this does not hold more generally because every positive real has two square roots, one positive and one negative. The notation $\sqrt x$ is used for the positive square root.

Square roots can also be written in exponent notation, so that $x^{\frac 12}$ is equal to the square root of $x$ . Note that this agrees with all the laws of exponentiation, properly interpreted. For example, $\left(x^{\frac12}\right)^2 = x^{\frac12 \cdot 2} = x^1 = x$ , which is exactly what we would have expected. This notion can also be extended to more general rational, real or complex powers, but some caution is warranted because these do not give functions. In particular, if we require that $x^{\frac 12}$ always gives the positive square root of a positive real number, then the equation $\left(x^2\right)^{\frac 12} = x$ does not hold. For example, replacing $x$ with $-2$ gives $2$ on the left but gives $-2$ on the right.

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Square root

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