Simplify square roots calculator (surds)

Learn the squares up to 144: 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144. Always use the largest one that divides in. Using 4 on $\sqrt{72}$ gives $2\sqrt{18}$, which isn't finished: $\sqrt{18}$ simplifies again to $3\sqrt{2}$, giving $6\sqrt{2}$.

• Adding: only like surds combine. $\sqrt{50} + \sqrt{8} = 5\sqrt{2} + 2\sqrt{2} = 7\sqrt{2}$. Simplify first to spot them.
• Multiplying: $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$, and $\sqrt{a} \times \sqrt{a} = a$. So $2\sqrt{3} \times 5\sqrt{3} = 10 \times 3 = 30$.

A fraction with a surd on the bottom is usually rewritten so the denominator is a whole number: multiply top and bottom by the surd. $\dfrac{6}{\sqrt{2}} = \dfrac{6\sqrt{2}}{2} = 3\sqrt{2}$. For denominators like $3 + \sqrt{2}$, multiply by $3 - \sqrt{2}$ (difference of two squares). Our guide to rationalising the denominator explains both types.

• Using a square factor that isn't the largest, and stopping too early.
• Adding unlike surds: $\sqrt{2} + \sqrt{3}$ is not $\sqrt{5}$.
• Writing $\sqrt{a + b} = \sqrt{a} + \sqrt{b}$ (it isn't).

Practise with our surds questions.

• "Give your answer in the form $a\sqrt{b}$" means simplest surd form, not a decimal.
• Surds appear in exact answers from Pythagoras and the quadratic formula.
• Surds are Higher tier at GCSE and Extended tier at IGCSE.

Surds keep answers exact. The diagonal of a 1 by 1 square is $\sqrt{2}$, not 1.414 (which is rounded). Exact answers appear in:

• Pythagoras: a right-angled triangle with sides 4 and 8 has hypotenuse $\sqrt{80} = 4\sqrt{5}$.
• The quadratic formula: $x^2 - 4x + 1 = 0$ gives $x = 2 \pm \sqrt{3}$.
• Trigonometry: $\sin 60° = \tfrac{\sqrt{3}}{2}$.

When a question says "give your answer in its simplest form" or "exact", keep the surd.