Math Core

A square garden has an area of 49 square feet. How long is each side? You need a number that, multiplied by itself, gives 49. Finding that number is called taking a square root, and it undoes squaring the same way subtraction undoes addition.

Squaring and unsquaring

Squaring 7 gives $7^2 = 49$ . Going backward, the square root of 49 is 7. The symbol for a square root is $\sqrt{\phantom{x}}$ , called a radical sign:

\sqrt{49} = 7 \quad \text{because} \quad 7^2 = 49

But $(-7)^2 = 49$ too. So 49 really has two square roots, 7 and $-7$ . The radical sign always means the positive one, called the principal square root. If you want the negative root, put a negative sign in front: $-\sqrt{49} = -7$ .

Definition

Square root

A square root of a number $n$ is a number that, when squared, equals $n$ . Every positive number has two square roots, one positive and one negative. The symbol $\sqrt{n}$ means the positive square root. Also, $\sqrt{0} = 0$ .

Perfect squares

A perfect square is the square of a whole number. Its square root is a whole number. These are worth knowing by heart:

$n$	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
$n^2$	1	4	9	16	25	36	49	64	81	100	121	144	169	196	225

Square roots of fractions work the same way: find the root of the top and the root of the bottom.

Worked example: Finding square roots

Evaluate each expression.

$\sqrt{144}$
$-\sqrt{81}$
$\sqrt{\dfrac{9}{25}}$
$\sqrt{0.36}$

Solutions.

$12^2 = 144$ , so $\sqrt{144} = 12$ .
$\sqrt{81} = 9$ , so $-\sqrt{81} = -9$ .
$\sqrt{9} = 3$ and $\sqrt{25} = 5$ , so $\sqrt{\dfrac{9}{25}} = \dfrac{3}{5}$ . Check: $\dfrac{3}{5} \cdot \dfrac{3}{5} = \dfrac{9}{25}$ .
$0.6 \cdot 0.6 = 0.36$ , so $\sqrt{0.36} = 0.6$ .

Common mistake

Taking a square root is not dividing by 2. $\sqrt{4} = 2$ matches $4 \div 2$ only by coincidence. Try another: $\sqrt{16} = 4$ , not 8, and $\sqrt{36} = 6$ , not 18. Always ask, "what number times itself gives this?" Also, a negative number like $-25$ has no square root among the numbers you know, since any number times itself is positive or zero.

Estimating square roots

Most numbers are not perfect squares. For example, 20 sits between the perfect squares 16 and 25, so $\sqrt{20}$ sits between $\sqrt{16} = 4$ and $\sqrt{25} = 5$ .

Since 20 is a little closer to 16 than to 25, $\sqrt{20}$ is a little closer to 4. In fact, $\sqrt{20} \approx 4.47$ . Numbers like $\sqrt{20}$ are irrational: their decimals go on forever without repeating, so you estimate them or leave them as $\sqrt{20}$ .

The square root of 20 is about 4.47, between 4 and 5.

Worked example: Estimating a square root

Between which two consecutive whole numbers is $\sqrt{58}$ ? Which one is it closer to?

Find the perfect squares on either side of 58: $7^2 = 49$ and $8^2 = 64$ .

49 < 58 < 64 \quad \Longrightarrow \quad 7 < \sqrt{58} < 8

58 is 9 away from 49 but only 6 away from 64, so $\sqrt{58}$ is closer to 8. (It is about 7.62.)

Solving equations with squares

To solve an equation like $x^2 = 64$ , ask which numbers square to 64. There are two: $x = 8$ or $x = -8$ . You can write this as $x = \pm 8$ , read "plus or minus 8."

Worked example: Solving an equation with a square

Solve $x^2 + 11 = 36$ .

Get $x^2$ by itself first, then take square roots:

\begin{aligned} x^2 + 11 &= 36 \\ x^2 &= 25 \\ x &= 5 \quad \text{or} \quad x = -5 \end{aligned}

Check: $5^2 + 11 = 36$ and $(-5)^2 + 11 = 36$ .

Cube roots

The cube root undoes cubing. Since $4^3 = 64$ , the cube root of 64 is 4, written $\sqrt[3]{64} = 4$ . Unlike square roots, a negative number does have a cube root: $\sqrt[3]{-8} = -2$ because $(-2)^3 = -8$ .

Roots undo powers

\sqrt{n} = r \text{ means } r^2 = n \text{ and } r \ge 0 \qquad\qquad \sqrt[3]{n} = r \text{ means } r^3 = n

Tip

To check any root, raise your answer to the power. If $\sqrt{196} = 14$ is right, then $14^2$ should be 196. It is: $14 \cdot 14 = 196$ .