Rationalize Denominator Calculator

Rationalize Denominator Calculator

Rationalize a denominator containing a square root and see the exact result written in a cleaner form.

Last updated: June 2026 | By Patchworkr Team

Calculation steps
Start with 1 / sqrt(2)
Multiply top and bottom by sqrt(2).
Result = 1 sqrt(2) / 2
No further numeric reduction is possible.
Rationalized form
1 sqrt(2) / 2
Exact structure
The result keeps the radical in the numerator and removes it from the denominator.

What Does Rationalizing The Denominator Mean?

Rationalizing the denominator means rewriting a fraction so the denominator no longer contains a radical. For a simple form like 1 / sqrt(n), multiply numerator and denominator by sqrt(n).

That turns the denominator into an integer while keeping the value of the expression unchanged.

How To Rationalize A Denominator

  1. Write the denominator as a square root.
  2. Multiply the numerator and denominator by the same radical.
  3. Simplify the square root in the denominator.
  4. Reduce any remaining numeric factor if possible.
a / sqrt(b) = a sqrt(b) / b

Worked Example

For 3 / sqrt(5), the rationalized form is 3 sqrt(5) / 5.

3 / sqrt(5)
= 3 sqrt(5) / 5

Frequently Asked Questions

Does this change the value of the expression?

No. Multiplying by sqrt(n)/sqrt(n) is multiplying by 1, so the value stays the same.

Can I use negative radicands?

No. The calculator requires a positive radicand for the square-root form used here.

Does the calculator simplify the numeric fraction too?

Yes. If the numerator and denominator share a common factor, it reduces them.

Can I enter decimals?

This calculator is designed for whole-number inputs so the exact rationalized form stays clear.

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