Rationalize the Denominator Calculator with Steps
Rationalize a / (b + d√c): multiply by √c or by the conjugate, simplify the radical and the fraction, and get the exact result with a decimal check.
How it works
a/(d√c) = a√c/(d·c) · a/(b + d√c) = a(b − d√c)/(b² − d²c) — the conjugate turns the denominator into a difference of squares
Rationalizing the denominator rewrites a fraction so no square root remains in the denominator, the standard form for final answers in algebra. For a single root, multiply top and bottom by that root, because √c · √c = c. For a binomial denominator b + d√c, multiply by its conjugate b − d√c: the product (b + d√c)(b − d√c) = b² − d²c is a whole number. The calculator also simplifies the radical first (√12 = 2√3) and cancels any common factor at the end.
Worked example
Rationalize 3/(2 + √5). Multiply top and bottom by the conjugate 2 − √5: the numerator is 3(2 − √5) = 6 − 3√5 and the denominator is 2² − 5 = −1. Dividing by −1 gives −6 + 3√5 ≈ 0.708204, the same value as the original fraction.
Sources: Rationalisation (mathematics) — Wikipedia · Rationalize the denominator — Math is Fun
Frequently asked questions
Why rationalize the denominator?
It is the conventional simplified form: it makes answers easy to compare, to add with other fractions and to estimate by hand. Historically it also made long division by an irrational number unnecessary.
What is a conjugate?
The conjugate of b + d√c is b − d√c (same terms, middle sign flipped). Their product b² − d²c has no root because the cross terms cancel.
How do I enter 5/(4√2)?
Set a = 5, b = 0, d = 4 and c = 2. The calculator multiplies by √2 and returns 5√2/8.
How do I enter a denominator like 3 − √7?
Use b = 3, d = −1, c = 7. A negative d means the root is subtracted.
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