Surds calculator
Simplify √72
6√2
Working
- Find the largest square number that divides 72: 36 (6²). √72 = √36 × √2 = 6√2.
Your result
6√2. As a decimal that is about 8.49, but the exact surd form is the answer GCSE questions ask for.
Surds, step by step
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Your mock plan, in three taps
A surd is simplified by taking out its largest square factor: √72 = √36 × √2 = 6√2, and √12 + √27 = 2√3 + 3√3 = 5√3.
Questions people ask
How do you simplify a surd?
Find the largest square number that is a factor of the number under the root, and take its square root outside. √48 = √16 × √3 = 4√3.
How do you rationalise the denominator?
Multiply the top and bottom by the surd on the bottom. 6/√3 × √3/√3 = 6√3/3 = 2√3. If the bottom is like 2 + √3, multiply by 2 − √3 instead.
Can you add √2 and √3?
No. √2 + √3 does not simplify because the numbers under the roots are different. Simplify each surd first: √8 + √18 = 2√2 + 3√2 = 5√2.
Why rationalise a denominator?
It is the agreed simplest form, so answers can be compared, and it is easier to work with a whole-number denominator.
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How we work this out
What it does
Simplify surds, add and subtract them, and rationalise denominators, with every step shown.
Whole numbers under the root up to 1,000,000; whole numbers in front from −999 to 999.
Method
- √(ab) = √a × √b: find the largest square number that divides the number under the root.
- Only like surds (the same number under the root) can be collected.
- a/√b: multiply top and bottom by √b. a/(b + √c): multiply top and bottom by b − √c, so the bottom becomes b² − c.