Rationalize Denominator Calculator
Remove radicals from denominators by multiplying with conjugates or powers, showing simplified results and LaTeX preview with quick examples automatically.
Equation Preview
Helping Notes
If the denominator is √a, multiply by √a/√a. For a ± b√c, multiply by its conjugate a ∓ b√c.
For nthRoot(a,n), multiply by nthRoot(a,n)n−1. The goal is removing radicals from the denominator only.
Results
Inputs Summary
Rationalized Form
Simplified Denominator
What Is a Rationalize Denominator Calculator?
A Rationalize Denominator Calculator rewrites expressions so the denominator contains no radicals or imaginary parts. Given inputs such as , , , or , it multiplies by an appropriate conjugate (or identity factor) to obtain an equivalent form with a rational denominator. The tool shows each step, simplifies radicals, combines like terms, and returns exact forms (with ) as well as optional decimals for quick checks.
About the Rationalize Denominator Calculator
Rationalization relies on identities that eliminate radicals or imaginary units from denominators. For a single square root, use . For binomials with one root, multiply by the conjugate: . For sums of two roots, use when . Higher roots use minimal polynomials: with cube roots, lets us clear . Complex denominators are handled by multiplying by the complex conjugate: . The calculator applies these patterns, simplifies factors, and preserves domain notes (e.g., , ).
Single root:
Binomial with one root:
Two roots:
nth root:
Cube‑root binomial:
Complex:
How to Use This Rationalize Denominator Calculator
- Enter your expression. The parser accepts radicals, exponents, variables, and complex numbers.
- Choose options: simplify radicals, combine like surds, and show exact or decimal forms.
- Compute. The tool selects the correct conjugate/identity factor, multiplies, and simplifies step‑by‑step.
- Copy the final rationalized result and, if needed, the derivation for assignments or reports.
Examples
- Single radical: .
- Binomial with root: .
- Two roots: .
- Cube‑root pair: .
- Complex: .
- Variable mix: , with domain .
Formula Snippets Ready for Rendering
FAQs
Why rationalize the denominator?
It yields cleaner arithmetic, easier comparison of fractions, and standard textbook forms that avoid radicals or imaginaries below the fraction bar.
Is rationalization always necessary?
No—it’s a convention. In numerical work, leaving radicals may be fine; in algebraic answers, rationalized form is typically preferred.
How do I handle even roots with variables?
Assume real principal roots and record domain conditions like to keep expressions real and denominators nonzero.
What about multiple different radicals?
Rationalize stepwise or use identities that match the minimal polynomial (e.g., the cube‑root identity) to clear radicals efficiently.
Can I rationalize complex denominators?
Yes—multiply by the complex conjugate so the denominator becomes , a real number.
Will rationalization change the value?
No. You multiply by 1 in a disguised form (conjugate over itself) and then simplify algebraically.
What if the denominator is already rational?
Then no change is needed; the calculator leaves it as is and may still simplify the numerator.