How to Simplify Square Roots (Surds) Step by Step

To simplify a square root, split the number into a perfect square times something else, then take the square root of the perfect square outside: $\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$. A surd is fully simplified when the number under the root has no square factors left.

The key rule

So if $a$ is a perfect square, $\sqrt{a}$ comes out as a whole number.

Method 1: Find the largest square factor

Perfect squares to look for: $4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144$.

$\sqrt{50}$: $50 = 25 \times 2$, so

$\sqrt{72}$: the largest square factor is $36$:

Method 2: Prime factorisation (always works)

Write the number as primes and pair them up. Each pair comes out as one factor.

Pairs: one pair of 2s and one pair of 3s come out as $2 \times 3 = 6$; one 2 stays inside. So $\sqrt{72} = 6\sqrt{2}$.

For cube roots, group factors in threes instead: $54 = 2 \times 3^3$, so $\sqrt[3]{54} = 3\sqrt[3]{2}$.

Adding and subtracting surds

Only like surds (same number under the root) can be combined, just like like terms in algebra:

Simplify first to reveal like surds: $\sqrt{8} + \sqrt{18} = 2\sqrt{2} + 3\sqrt{2} = 5\sqrt{2}$.

Multiplying surds

Rationalising the denominator

To remove a root from the bottom of a fraction, multiply top and bottom by that root:

Common mistakes

• $\sqrt{a + b} \ne \sqrt{a} + \sqrt{b}$. For example, $\sqrt{9 + 16} = 5$, but $3 + 4 = 7$.
• Not using the largest square factor.
• Adding unlike surds: $\sqrt{2} + \sqrt{3}$ cannot be simplified.

Practice questions

1. Simplify $\sqrt{48}$.
2. Simplify $\sqrt{200}$.
3. Simplify $\sqrt{12} + \sqrt{27}$.

Answers: 1) $4\sqrt{3}$ 2) $10\sqrt{2}$ 3) $5\sqrt{3}$

The square root calculator shows the prime factorisation and the pairs, with an exact and decimal answer. The cube root calculator does the same in threes. Surds often appear in the Pythagorean theorem.

Frequently asked questions

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