Rationalise the Denominator Calculator

This calculator rationalises a fraction of the form a over root b, removing the square root from the denominator so the expression is easier to work with, add or compare. You enter the numerator, a, and the number under the root, b, and the calculator multiplies the top and bottom by root b, which turns the denominator into a whole number, then simplifies the surd and cancels any common factor between the numerator and denominator. The result card shows the simplified fraction in the form a over root b, the decimal value of the original expression so you can check the rationalised version matches, and the simplified radicand left under the root. This matters because a surd sitting in the denominator is awkward in algebra: it is harder to add fractions with different surd denominators, harder to compare sizes, and harder to spot when two expressions are actually equal. Rationalising fixes that without changing the value, since you are effectively multiplying by a form of 1. Use it to check homework, confirm a simplification step, or convert a surd expression into the tidy form usually expected in algebra and calculus courses. Try entering 1 and 2 to see 1 over root 2 become root 2 over 2, about 0.707107, then try 2 and 8 to see the same simplified answer appear from a different starting fraction.

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Numerator a

Radicand b (under the root)

a/√b = √2 / 2

Decimal value0.707107

Radicand2

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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