Lesson 2-2: Roots

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.

Property

The principal square root is the positive square root of a number. Any positive number $x$ has both a positive and negative square root, which can be written as $\pm \sqrt{x}$.

Examples

• $\sqrt{64} = 8$
• $-\sqrt{9} = -3$
• $\sqrt{\frac{4}{25}} = \frac{2}{5}$

Explanation

Every positive number has two dance partners for its square root, one positive and one negative! But the radical symbol $\sqrt{}$ is a bit picky and only wants to dance with the positive partner, which we call the 'principal square root.' So, unless you see a grumpy negative sign outside, always choose the happy, positive root!

Property

Taking a square root is the opposite of squaring a number.
To solve an equation of the form $x^2 = k$ (where $k > 0$), we take the square root of both sides.
Because a positive number has two square roots, the solution is written as:

Examples

• To solve the equation $x^2 = 81$, we take the square root of both sides. The solutions are $x = \pm\sqrt{81}$, which means $x = 9$ and $x = -9$.