Rationalise The Denominator Calculator

Work out rationalise the denominator instantly with clear inputs, formula shown and shareable results.

About Rationalise The Denominator Calculator

Multiplying top and bottom by the conjugate b − c√d clears the surd, because (b + c√d)(b − c√d) = b² − c²d is rational. The result splits into a rational part plus a multiple of √d.

Formula

Rationalising

a/(b + c√d) = a(b − c√d) / (b² − c²d)

Worked example

With the default values below, Rationalise The Denominator Calculator returns Value of the fraction of 1.35924552. Change any input above and every figure here is recalculated in your browser.

Inputs used

Numerator a: 6
Rational part of the denominator b: 3
Surd coefficient c: 1
Surd radicand dminimum 0: 2

Results

Value of the fraction: 1.35924552
Rational part ab/(b² − c²d): 2.57142857
Coefficient of √d: -0.85714286
b² − c²d: 7.00000000

Step by step

1Values used
2Rationalising
3Value of the fraction= 1.35924552
4Rational part ab/(b² − c²d)= 2.57142857
5Coefficient of √d= -0.85714286
6b² − c²d= 7.00000000

Assumptions and limits

The result is derived only from the 4 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

Rationalise 6/(3 + √2).

Multiply by (3 − √2)/(3 − √2) to get 6(3 − √2)/7, that is 18/7 − (6/7)√2 ≈ 1.3592455.

Why bother rationalising?

It makes fractions easier to compare and to add, and before calculators it made hand computation far simpler.