A simplified answer traditionally has no radicals in the denominator. Rationalising makes expressions easier to compare and add, and was essential when calculations were done by hand with tables.
How to use the Rationalize Denominator Calculator
- Expression
- a
- b
- c (under the root)
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
For a single radical the numerator and denominator are multiplied by √c. For a binomial denominator they are multiplied by its conjugate, using (b + √c)(b − √c) = b² − c. The result is simplified.
Formula
a/(b√c) = a√c / (bc)
a/(b + √c) = a(b − √c) / (b² − c)
Worked example
Inputs — Expression: a / (b + √c); a: 3; b: 2; c (under the root): 3.
Results — Rationalized form = 6 − 3√3; Decimal value = 0.8038475773; Steps = Multiply top and bottom by the conjugate 2 − √3.
The denominator becomes 2² − 3 = 1..
Key terms
- Conjugate
- The same binomial with the sign between the terms changed.
Frequently asked questions
Why rationalise a denominator?
It is the conventional simplified form and makes sums of fractions easier.
What is a conjugate?
b − √c is the conjugate of b + √c; their product has no root.
How do I rationalise 1/√2?
Multiply by √2/√2 to get √2/2.
Does rationalising change the value?
No – you multiply by a form of 1.