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Principal Value of a Square Root

A square root of a number $b$ is the solution of the equation $x^{2} = b$ . It is a number that when multiplied by itself gives you $b$ . Every positive number $b$ has two square roots , denoted $\sqrt{b}$ and $- \sqrt{b}$ . The principal square root of $b$ is the positive square root, denoted $\sqrt{b}$ .

Example 1:

The square roots of $25$ are $\sqrt{25} = 5$ and $- \sqrt{25} = - 5$ since $5^{2} = 25$ and ${(- 5)}^{2} = 25$ .

The principal square root of $25$ is $\sqrt{25} = 5$ .

Example 2:

Find the real roots of the equation $x^{2} = 100$ .

$x = \pm \sqrt{100} = \pm 10$

Therefore, the roots are $10$ and $- 10$ .

Example 3:

Simplify $\sqrt{36}$ .

$\sqrt{36}$ indicates the principal (or positive) square root so $\sqrt{36} = 6$ .