Rationalise the Denominator Calculator

This rationalise the denominator calculator removes a square root from the bottom of a fraction of the form a over root b. Enter the numerator a and the number under the root, b, and the calculator multiplies top and bottom by root b to give an equivalent fraction with a whole number denominator, then simplifies it. Rationalising the denominator is a standard tidy-up step in algebra, because a surd in the denominator is awkward to work with and harder to add or compare. The decimal value is shown so you can check the two forms match. The method, a worked example and the assumptions are below.

Numerator a

Radicand b (under the root)

Decimal value

Radicand

The formula

Multiply numerator and denominator by root b: a / root b = (a root b) / b. Then simplify the surd and cancel any common factor between the numerator coefficient and the denominator.

Worked example

1 / root 2 becomes (1 times root 2) / 2 = root 2 / 2, about 0.707107. And 2 / root 8 simplifies to root 2 / 2 as well. Enter 1 and 2 to confirm.

Frequently asked questions

Why rationalise the denominator?

A surd in the denominator is hard to work with. Rewriting it with a whole number denominator makes the expression easier to add, compare and evaluate.

How does it work?

Multiply the top and bottom by the root in the denominator, which turns root b times root b into b, a whole number.

Does the value change?

No. You are multiplying by a form of 1, so the value is unchanged, only the way it is written.

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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