Square Roots and Their Properties

Square Roots

Definition and Properties

The square root of a number is a value that, when multiplied by itself, yields the original number. For any positive real number, there are always two square roots: one positive (the principal root) and one negative.

• Principal Root: The positive square root of a number. For example, the principal root of 9 is 3.

• Negative Root: The negative square root of a number. For example, the negative root of 9 is -3.

• General Principle: If $x^2 = a$ and $a > 0$, then $x = \pm \sqrt{a}$.

Examples

• Example 1: Find the square roots of 9.

  • Test: $3^2 = 9$ and $(-3)^2 = 9$

  • Therefore, the square roots of 9 are 3 and -3.

• Example 2: Find the square roots of 16.

  • Test: $4^2 = 16$ and $(-4)^2 = 16$

  • Therefore, the square roots of 16 are 4 and -4.

Key Formula

• For any positive real number $a$:

  • $x^2 = a \implies x = \pm \sqrt{a}$

Additional info:

• Square roots are fundamental in algebra and are used in solving quadratic equations, simplifying expressions, and understanding the properties of exponents.

• For negative numbers, the square root is not a real number; instead, it is an imaginary number (e.g., $\sqrt{-1} = i$).