Lesson 1: From Squares to 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

For any nonnegative number $n \geq 0$:

Examples

Property

If $a$ is a nonnegative real number, then $(\sqrt{a})^2 = a$.

Examples

• To simplify $(\sqrt{7})^2$, the square and square root cancel, leaving $7$.
• For $(-\sqrt{13})^2$, the expression means $(-\sqrt{13})(-\sqrt{13})$. The negatives cancel, and $\sqrt{13} \cdot \sqrt{13} = 13$.
• To simplify $(5\sqrt{3})^2$, you square both the coefficient and the radical: $5^2 \cdot (\sqrt{3})^2 = 25 \cdot 3 = 75$.

Explanation

Squaring a square root is like undoing an operation. The square and the square root cancel each other out, leaving only the number that was inside the radical. This is because $(\sqrt{a})^2$ means $\sqrt{a} \cdot \sqrt{a} = \sqrt{a^2} = a$.