Math July 13, 2026 · 8 Min Read

Rationalize Denominator Calculator – Guide & Formulas

Rationalize denominators with square roots, cube roots, and binomial radicals. See step-by-step multiplication to eliminate radicals from fractions.

Rationalize any denominator with our free online Rationalize Denominator Calculator. Enter fractions with radicals to see step-by-step multiplication that eliminates radicals from the bottom.

Key Takeaway

Use the free Rationalize Denominator Calculator to rationalize denominators with square roots, cube roots, and binomial radicals. see step-by-step multiplication to eliminate radicals from fractions. Get instant results with step-by-step explanations.

How to Use the Rationalize Denominator Calculator

  1. Step 1: Enter the numerator of your fraction
  2. Step 2: Enter the denominator (with or without radicals)
  3. Step 3: Click Calculate to see the rationalized form
  4. Step 4: Review the step-by-step conjugate multiplication process

The Formula

To rationalize a simple radical denominator: multiply numerator and denominator by the radical. For binomial denominators like (a + √b), multiply by the conjugate (a - √b). The difference of squares eliminates the radical.

Variable Definitions

  • a: The numerator of the fraction
  • √b: The radical in the denominator to be eliminated
  • Conjugate: The expression with the opposite sign between terms (a + √b becomes a - √b)
  • Difference of Squares: The identity (a+b)(a-b) = a² - b² used to eliminate radicals

Rationalizing 5 / (3 + √2)

Eliminate the radical from the denominator of 5/(3 + √2).

  1. Step 1: Identify the conjugate of (3 + √2) as (3 - √2)
  2. Step 2: Multiply numerator and denominator by the conjugate: 5(3 - √2) / [(3 + √2)(3 - √2)]
  3. Step 3: Expand denominator using difference of squares: 3² - (√2)² = 9 - 2 = 7
  4. Step 4: Result = 5(3 - √2) / 7 = (15 - 5√2) / 7

Frequently Asked Questions

What does it mean to rationalize a denominator?

Rationalizing means rewriting a fraction so the denominator contains no radicals (square roots, cube roots, etc.). This is done by multiplying both numerator and denominator by an appropriate expression.

Why do we rationalize denominators?

Rationalized denominators are standard mathematical convention, easier to compute with manually, and required in many textbooks and standardized tests.

What is a conjugate?

The conjugate of (a + √b) is (a - √b). When multiplied together, the result is a² - b, which is a rational number with no radical.

How do I rationalize a cube root denominator?

For ∛b, multiply numerator and denominator by ∛(b²) so the denominator becomes ∛(b³) = b. For example, 1/∛3 becomes ∛9/3.

Can I rationalize denominators with variables?

Yes, the same principles apply. For example, 1/√x becomes √x/x after multiplying by √x/√x.