A surd is a root that cannot be simplified to a whole number or fraction, such as √2 or [3]5. Leaving it as a root keeps the value exact.

Square Roots and Surds: Simplifying Radicals Step by Step

Q: Is the square root of 9 equal to 3 or ±3?

√9 = 3. The root symbol means the non-negative root. It is the equation x^2 = 9 that has two solutions, 3 and -3.

What a square root is, how to estimate one, why $\sqrt{9}$ is $3$ and not $\pm 3$ , and how to simplify surds by pulling out perfect squares.

3 min read Supports 4 lessons

A square with area $49$ has sides of length $7$ . Finding that side is what a square root does: it undoes squaring.

7^2 = 49 \qquad\Longleftrightarrow\qquad \sqrt{49} = 7

$y = \sqrt{x}$ grows ever more slowly. It only exists for $x \ge 0$ .

Perfect squares and estimating

Numbers like $1, 4, 9, 16, 25, 36, \dots$ are perfect squares: their square roots are whole numbers. For anything else, find the perfect squares on either side.

Worked example

Estimate $\sqrt{50}$ .

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$49 < 50 < 64$ , so $7 < \sqrt{50} < 8$ .

Tip

Learn the perfect squares up to $15^2 = 225$ . They make estimating and simplifying much quicker.

One root or two?

Watch out

$\sqrt{9} = 3$ , not $\pm 3$ . The symbol $\sqrt{\;}$ always means the principal, non-negative root. But the equation $x^2 = 9$ has two solutions, $x = 3$ and $x = -3$ , because both square to $9$ .

You cannot take the square root of a negative number and get a real answer: no real number squares to something negative.

Surds

Most square roots are not whole numbers. $\sqrt{2} = 1.41421356\dots$ goes on for ever without repeating, so it cannot be written exactly as a decimal or fraction. Roots like this are called surds, and mathematicians usually leave them in root form because that is exact.

Product rule for roots

\sqrt{ab} = \sqrt{a}\cdot\sqrt{b} \qquad \sqrt{\frac{a}{b}} = \frac{\sqrt a}{\sqrt b}

Simplifying surds

To simplify a surd, find the largest perfect square that divides the number under the root, split it off, and take its root outside.

$\sqrt{72}$
Look for the largest perfect square factor of 72.

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A prime factorisation helps with larger numbers: pair up equal primes, and each pair comes out of the root as one copy. $\sqrt{360} = \sqrt{2^3 \cdot 3^2 \cdot 5} = 2 \cdot 3\sqrt{2 \cdot 5} = 6\sqrt{10}$ .

Adding surds

Surds add like algebraic like terms. $3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2}$ , just as $3x + 5x = 8x$ . Different surds may combine once simplified:

Worked example

Simplify $\sqrt{12} + \sqrt{27}$ .

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Simplify each.
$2\sqrt3 + 3\sqrt3$

Watch out

$\sqrt{a + b} \neq \sqrt a + \sqrt b$ . Check: $\sqrt{9 + 16} = \sqrt{25} = 5$ , but $\sqrt9 + \sqrt{16} = 7$ .

Other roots

A cube root undoes cubing: $\sqrt[3]{27} = 3$ because $3^3 = 27$ . Unlike square roots, cube roots of negative numbers are fine: $\sqrt[3]{-8} = -2$ . In general the $n$ th root $\sqrt[n]{a}$ can also be written as the power $a^{1/n}$ .

Common questions

Is the square root of 9 equal to 3 or ±3?: $\sqrt{9} = 3$ . The root symbol means the non-negative root. It is the equation $x^2 = 9$ that has two solutions, $3$ and $-3$ .
What is a surd?: A surd is a root that cannot be simplified to a whole number or fraction, such as $\sqrt{2}$ or $\sqrt[3]{5}$ . Leaving it as a root keeps the value exact.

Practise square roots and surds

The Square Roots lesson has worked examples and unlimited practice that adapts to you.

Open the lesson

Common questions

Is the square root of 9 equal to 3 or ±3?

$\sqrt{9} = 3$ . The root symbol means the non-negative root. It is the equation $x^2 = 9$ that has two solutions, $3$ and $-3$ .

What is a surd?

A surd is a root that cannot be simplified to a whole number or fraction, such as $\sqrt{2}$ or $\sqrt[3]{5}$ . Leaving it as a root keeps the value exact.