Principal Square Root

Definition

Every positive number "n" has two square roots. One of them is positive ( or ) , and the other is negative ( ). Together, these are denoted as .

The principal square root is the positive number square root.

Unless otherwise stated, "the square root" of a number refers **ONLY** to the principal square root.

Important Consequence

The square root of n^{2} is the absolute value of n.

This is a compact, consise way of expressing the following, (using the number 25 as an example):

Example: Square Roots (Simple Equations)

When solving a simple equation such as x^{2} = 25, it must be observed that there are **two** solutions. The two solutions to this equation are 5 and -5, since both 5^{2} = 25 and (-5)^{2} = 25.

The solution above shows all of the steps.

When someone is proficient with square roots, usually only the steps shown below are written out.

Example: Square Roots (The Pythagorean Theorem)

When working with equations involving the Pythagorean Theorem, it is important to remember that the lengths of the sides of the triangle are positive numbers.