Square Roots: What They Mean and How to Estimate Them

Undoing a square

Squaring $5$ gives $25$. A square root runs that backwards:

$$5^2 = 25 \qquad\Longleftrightarrow\qquad \sqrt{25} = 5$$

Read $\sqrt{25}$ as “the number that multiplies by itself to give $25$”.

The name is literal. A square with area $25$ has sides of $5$, so taking the square root of an area gives you the side.

$$\text{area} = 5 \times 5 = 25 \qquad \text{side} = \sqrt{25} = 5$$

That is the whole idea: squaring goes from side to area, and a square root goes from area back to side.

Perfect squares

A perfect square is a whole number that some whole number squares to. These are worth recognising on sight, because a root that lands on one is exact.

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

Why the radical sign gives only the positive root

Both $5$ and $-5$ square to $25$, so the question “what squares to 25?” has two answers. The symbol $\sqrt{\phantom{x}}$ picks one of them:

$$\sqrt{25} = 5 \qquad \text{(the positive root, always)}$$

That is a definition, not a discovery. A symbol has to name one number to be usable — if $\sqrt{25}$ could mean either $5$ or $-5$, then $\sqrt{25} + 1$ would have no single value, and no formula containing a root would give a definite answer.

The second root has not disappeared. It is written in when it is wanted:

$$x^2 = 25 \quad\Rightarrow\quad x = \pm\sqrt{25} = \pm 5$$

Expression	Meaning	Answer
$\sqrt{25}$	the positive root	$5$
$-\sqrt{25}$	the negative root	$-5$
$x^2 = 25$	an equation, both roots	$5$ and $-5$

This is exactly why the quadratic formula carries a $\pm$ in front of its radical. The $\pm$ is doing work the $\sqrt{\phantom{x}}$ deliberately does not.

Why square roots keep turning up

Every area formula that involves a squared length runs backwards through a root. Give a square garden an area and ask for its fencing; give a circle an area and ask for its radius; and a root is how you get there.

The largest use is the Pythagorean theorem, which gives $c^2$ rather than $c$:

$$3^2 + 4^2 = c^2 \quad\Rightarrow\quad c^2 = 25 \quad\Rightarrow\quad c = \sqrt{25} = 5$$

Without the last step the answer is an area, not a length. Distance on a coordinate grid, the diagonal of a screen, and the reach of a ladder against a wall are all this same final square root.

Roots are also where irrational numbers first appear in school mathematics. $\sqrt{2}$ cannot be written as a fraction and its decimal never repeats, yet it is the exact diagonal of a $1 \times 1$ square — a length you can draw but not write down in full.

Roots that fall between whole numbers

Most numbers are not perfect squares, and their roots sit between two whole numbers. Trap the number between the perfect squares either side:

$$49 < 50 < 64 \quad\Rightarrow\quad 7 < \sqrt{50} < 8$$

$50$ is barely above $49$, so $\sqrt{50}$ is barely above $7$ — not halfway to $8$.

Squaring a guess checks it and improves it:

$$7.1^2 = 50.41 \quad \text{(slightly too big)}$$ $$7.07^2 = 49.98 \quad \text{(a little under)}$$

So $\sqrt{50} \approx 7.07$. Squaring your estimate is the check, and it works without a calculator or a table.

Cube roots

The same idea with three factors instead of two:

$$\sqrt[3]{27} = 3 \quad \text{because} \quad 3 \times 3 \times 3 = 27$$

The small $3$ in the notch says how many factors. A cube root undoes cubing, so it turns the volume of a cube back into its edge — the three-dimensional version of area back into side.

Cube roots of negatives do exist: $\sqrt[3]{-8} = -2$, because $(-2)^3 = -8$. An odd number of negative factors keeps the answer negative, which is the same sign rule as any other odd power.

Worked examples

Common mistakes

Practice problems