Math Core

A square rug has an area of $49$ square feet. How long is each side? You need a number that, multiplied by itself, gives $49$ . Undoing a square (or a cube) is what square roots and cube roots are for.

Square roots

Squaring a number means multiplying it by itself: $7^2 = 7 \cdot 7 = 49$ . A square root goes the other way.

Definition

Square root

A square root of a number $p$ is a number that gives $p$ when squared. The symbol $\sqrt{p}$ means the positive square root (also called the principal square root).

Every positive number has two square roots, one positive and one negative, because a negative times a negative is positive:

$7^2 = 49 \qquad \text{and} \qquad (-7)^2 = 49$

The symbol $\sqrt{49}$ means only the positive one, $7$ . If you want the negative one, write $-\sqrt{49} = -7$ . Also, $\sqrt{0} = 0$ .

Numbers like $49$ that are squares of whole numbers are called perfect squares. It is worth memorizing the first fifteen:

$n$	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. Since $\dfrac{3}{5} \cdot \dfrac{3}{5} = \dfrac{9}{25}$ , we have $\sqrt{\dfrac{9}{25}} = \dfrac{3}{5}$ .

Solving $x^2 = p$

The equation $x^2 = 49$ asks, "What numbers square to $49$ ?" There are two answers, so you write both:

$x^2 = 49 \quad\Longrightarrow\quad x = \sqrt{49} \ \text{ or } \ x = -\sqrt{49} \quad\Longrightarrow\quad x = 7 \ \text{ or } \ x = -7$

A short way to write this is $x = \pm 7$ ("plus or minus $7$ ").

Common mistake

Two different questions, two different answers:

"What is $\sqrt{49}$ ?" has one answer: $7$ .
"Solve $x^2 = 49$ " has two answers: $7$ and $-7$ .

Also, $x^2 = -49$ has no real solution, because no real number squared is negative.

Cube roots

Cubing a number means using it as a factor three times: $4^3 = 4 \cdot 4 \cdot 4 = 64$ . The volume of a cube with side $4$ is $64$ cubic units.

Definition

Cube root

The cube root of $p$ , written $\sqrt[3]{p}$ , is the number that gives $p$ when cubed. So $\sqrt[3]{64} = 4$ because $4^3 = 64$ .

Cube roots are different from square roots in two ways:

Every number has exactly one real cube root.
The cube root of a negative number is negative: $(-4)^3 = (-4)(-4)(-4) = -64$ , so $\sqrt[3]{-64} = -4$ .

So $x^3 = 64$ has just one solution, $x = 4$ , and $x^3 = -64$ has just one solution, $x = -4$ .

The perfect cubes up to $10^3$ :

$n$	1	2	3	4	5	6	7	8	9	10
$n^3$	1	8	27	64	125	216	343	512	729	1000

Square roots vs. cube roots

$x^2 = p$ (with $p > 0$ ) has two solutions: $x = \pm\sqrt{p}$ .
$x^3 = p$ has one solution: $x = \sqrt[3]{p}$ , for any $p$ , positive or negative.
If $p$ is not a perfect square, $\sqrt{p}$ is irrational. If $p$ is not a perfect cube, $\sqrt[3]{p}$ is irrational.

Worked example: Evaluating roots

$\sqrt{144}$
$\sqrt[3]{125}$
$\sqrt[3]{-8}$
$\sqrt{\dfrac{16}{81}}$
$\sqrt[3]{\dfrac{1}{27}}$

Solutions.

$12^2 = 144$ , so $\sqrt{144} = 12$ .
$5^3 = 125$ , so $\sqrt[3]{125} = 5$ .
$(-2)^3 = -8$ , so $\sqrt[3]{-8} = -2$ .
$\sqrt{\dfrac{16}{81}} = \dfrac{4}{9}$ , because $4^2 = 16$ and $9^2 = 81$ .
$\sqrt[3]{\dfrac{1}{27}} = \dfrac{1}{3}$ , because $\left(\dfrac{1}{3}\right)^3 = \dfrac{1}{27}$ .

Worked example: Solving equations

Solve each equation.

$x^2 = 100$
$x^3 = 343$
$x^2 = 17$

Solutions.

$x = \pm\sqrt{100} = \pm 10$ . Both $10$ and $-10$ work.
$x = \sqrt[3]{343} = 7$ , since $7 \cdot 7 \cdot 7 = 343$ . There is only one solution.
$17$ is not a perfect square, so the answers are left as roots: $x = \sqrt{17}$ or $x = -\sqrt{17}$ . Both are irrational.

Worked example: A real-world cube

A cube-shaped box holds $216$ cubic inches. How long is each edge?

The volume of a cube is $s^3$ , so $s^3 = 216$ . Then $s = \sqrt[3]{216} = 6$ inches, because $6 \cdot 6 \cdot 6 = 216$ . (A length can't be negative, and a cube root has only one value anyway.)

Tip

Check any root by undoing it. If you think $\sqrt[3]{512} = 8$ , cube it: $8 \cdot 8 = 64$ and $64 \cdot 8 = 512$ . It checks.