Surds Simplification Calculator

This surds simplification calculator takes the square root of a whole number and writes it in its simplest surd form, pulling out the largest perfect square factor. Enter a number and the calculator gives the simplified root, such as the square root of 72 equals 6 root 2, along with the decimal value. A surd is a root that cannot be reduced to a whole number, and simplifying it makes expressions tidier and exact, which is preferred over decimals in algebra and trigonometry. If the number is a perfect square the result is a whole number. The method, a worked example and the assumptions are below.

Number under the root

Decimal value

As a times root b

The formula

Factor the number under the root into a perfect square times a remainder: n = a squared times b, where b has no square factors. Then the square root of n equals a times the square root of b.

Worked example

72 = 36 times 2, and 36 is a perfect square, so the square root of 72 is 6 root 2, about 8.485281. Enter 72 to confirm.

Frequently asked questions

What is a surd?

An irrational root, such as the square root of 2, that cannot be written exactly as a fraction or whole number.

How do I simplify a square root?

Take out the largest perfect square factor. The square root of 50 is 5 root 2 because 50 = 25 times 2.

What if the number is a perfect square?

Then the root is a whole number and there is no surd left, for example the square root of 49 is 7.

Who this calculator is for

This calculator is for students, programmers and anyone needing a quick, reliable result.

What this calculator assumes

You enter valid input.
The standard algorithm is applied.
Results are rounded for display.

Formula and sources

Related calculators

Scientific Calculator: general scientific maths.
Percentage Calculator: percentages and changes.
Prime Factorisation Calculator: factor a number into primes.

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