Surd simplifier and denominator rationaliser
Non-calculator papers demand exact answers, which means surds must be fully simplified and denominators rationalised. Enter a number under the root to see the largest square factor pulled out, or enter a fraction such as 6/√3 to see the rationalising step written out in full.
Simplify k√n (and rationalise if needed)
6√2
Decimal value: 8.485281
Working
- √72 = √(36 × 2) = 6√2
The method behind it
- Split the number into its largest square factor: √(k² × m) = k√m.
- √a × √b = √(ab) and √a ÷ √b = √(a/b), but √a + √b ≠ √(a + b).
- Rationalise a/√b by multiplying top and bottom by √b to get a√b / b.
How to use this tool
1. Enter the number under the root
Type the value inside the square root, for example 72, and an optional coefficient in front of it.
2. Read the simplified surd
The tool finds the largest square factor and shows the k√m form, plus the decimal value for checking.
3. Rationalise if there is a denominator
Enter a denominator to see the fraction multiplied by √b over √b and simplified to an exact answer.
Frequently asked questions
How do you simplify a surd?
Find the largest square number that divides the value under the root and take its square root outside. √72 = √(36 × 2) = 6√2. Using a smaller square factor such as 4 still works but needs a second simplification step.
What does rationalising the denominator mean?
It means removing the surd from the bottom of a fraction by multiplying the top and bottom by that surd. 6/√3 becomes 6√3/3 = 2√3, which is the form mark schemes expect.
Is √a + √b the same as √(a + b)?
No. √9 + √16 = 3 + 4 = 7, while √25 = 5. Square roots can be multiplied and divided term by term, but never added or subtracted that way.
When do I have to leave an answer in surd form?
Whenever the question says 'give your answer in exact form', 'in surd form', or the paper is non-calculator. A rounded decimal usually loses the accuracy mark even when the method is correct.
Practise this properly, not just once
A solver gives you the answer; exam marks come from the working. MathsGradeUp pairs exam-board questions with step-by-step feedback and revision notes on the exact topics you keep dropping marks on.