Section 7.1: Finding Square Roots

Property

A number $s$ is called a square root of $N$ if $s^2 = N$. We use a special symbol called a radical sign, $\sqrt{\hphantom{0}}$, to denote the positive square root of a number. For example, $\sqrt{16}$ means "the positive square root of 16," so $\sqrt{16} = 4$. Numbers such as 16 and 25 are called perfect squares because they are the squares of whole numbers.

Examples

• 4 is a square root of 16 because $4^2 = 16$.
• 9 is a square root of 81 because $9^2 = 81$.
• $\frac{3}{5}$ is a square root of $\frac{9}{25}$ because $(\frac{3}{5})^2 = \frac{9}{25}$.

Explanation

Think of a square root as the reverse of squaring a number. If you know the area of a square, the square root tells you the length of its side. It answers the question: "What number, when multiplied by itself, gives this result?"

Property

The number $s$ is called a square root of a number $b$ if $s^2 = b$. Every positive number has two square roots, one positive and one negative.

Examples

• The two square roots of 49 are 7 and -7, because $7^2 = 49$ and $(-7)^2 = 49$.
• A square garden has an area of 64 square feet. Its side length is a square root of 64, which is 8 feet.
• Since $12^2 = 144$, we know that 12 is a square root of 144. The other square root is $-12$.

Explanation

Finding a square root is the reverse of squaring a number, like finding a square's side from its area.