Rationalize Denominator Calculator - Conjugate Method Step-by-Step

A rationalize denominator calculator rewrites a fraction so that no radical sits in the bottom, by multiplying the fraction by a form of 1 chosen to clear square roots from the denominator.

• • Algebra homework: Rewrite answers like 1/sqrt(3) and 1/(sqrt(2) + sqrt(5)) into the form a*sqrt(b)/c that most textbooks expect.
• • Calculus limits: Clear radicals from the denominator of expressions like (1 - cos x)/sin x so l'Hopital or trig identities apply cleanly.
• • Physics and engineering: Rescale quantities with square roots in the denominator, such as effective resistance formulas.
• • Symbolic math checks: Verify a hand-derived rationalized form against a calculator result before using the expression in a longer derivation.

Rationalizing the denominator multiplies a fraction by a form of 1 so the bottom becomes a rational number while the value does not change. The classic tool is the conjugate of the denominator, the same expression with the sign between its two terms reversed.

For a single-term denominator a*sqrt(b), multiply by sqrt(b)/sqrt(b) so the bottom becomes a*b. For a binomial a*sqrt(b) +/- c*sqrt(d), the conjugate is a*sqrt(b) -/+ c*sqrt(d), and multiplying by the conjugate swaps the radical for a^2*b - c^2*d.

Once the denominator is rational, our Fraction Calculator helps combine the result with other fractions over a common denominator.

Write the denominator as one or two terms of the form a*sqrt(b). For a single term the conjugate is sqrt(b); for two terms it is the same expression with the sign flipped. Multiplying by the conjugate uses (x + y)(x - y) = x^2 - y^2 to swap the radical for a rational number.

• a, b: Coefficient and radicand of the numerator term a*sqrt(b). Use b = 1 for no radical.
• c, d: Coefficient and radicand of the first denominator term c*sqrt(d). Use d = 1 for no radical.
• op: Sign between the two denominator terms. The conjugate flips this sign.
• e, f: Coefficient and radicand of the second denominator term e*sqrt(f). Set e = 0 for the single-term case.

The new denominator is c^2*d - e^2*f for the binomial case, or c*d for the single-term case, always a rational integer. The new numerator is built by distributing the original numerator across the conjugate.

After multiplication, the result is simplified by dividing both the numerator coefficients and the new denominator by their greatest common divisor, keeping the fraction in lowest terms.

According to Khan Academy, multiplying by the conjugate uses the difference-of-squares identity a^2 - b^2 = (a+b)(a-b) to clear radicals from a binomial denominator in one step.

After rationalizing, the result is often combined with another fraction by multiplication, and our Multiplying Fractions Calculator handles that next step in the same a/b input form.

Four ideas drive every rationalize-the-denominator calculation. Knowing them makes it easier to read the calculator's output and check that the work is correct.

The same identity covers the single-term case with b set to zero: (a)(a) = a^2. Both branches share the same machinery, so a single calculator handles every rationalize-the-denominator problem that uses a square-root radical.

The rationalized form is an equivalent fraction of the input, so the Equivalent Fractions Calculator confirms the value has not changed.

The form mirrors the structure of a symbolic fraction. Type the numerator term, the first denominator term, pick the sign between two denominator terms, and add an optional second denominator term.

1. 1 Type the numerator coefficient a: Enter the integer coefficient in front of the radical in the numerator. Use 1 when the numerator is a single radical like sqrt(2).
2. 2 Type the numerator radicand b: Enter the radicand under the radical. Use 1 to mean no radical, so the numerator is just the integer a.
3. 3 Type the first denominator coefficient c: Enter the integer coefficient of the first denominator term c*sqrt(den1B).
4. 4 Type the first denominator radicand d: Enter the radicand of the first denominator term. Use 1 for a plain integer c and a positive non-square integer for sqrt(d).
5. 5 Pick the sign between denominator terms: Choose + or - from the operator menu. The calculator will use the opposite sign for the conjugate.
6. 6 Add the second denominator term or skip it: Set the second coefficient to 0 for a single-term denominator. Otherwise enter the second coefficient and radicand for a binomial like 1/(sqrt(3) + sqrt(5)).

Try a = 1, b = 1, c = 1, d = 3, op = +, e = 1, f = 5 to see 1/(sqrt(3) + sqrt(5)) rationalized step by step. The result panel shows the original form, the conjugate, the intermediate, and the simplified (sqrt(5) - sqrt(3))/2.

For the decimal value of the rationalized form, use the Root Calculator on any radicand that can be simplified.

A focused rationalize-the-denominator tool handles both branches in one place, with the work shown symbolically so each step can be checked.

• • Handles single-term and binomial cases: Works for 1/(a*sqrt(b)) and 1/(a*sqrt(b) +/- c*sqrt(d)) without switching forms.
• • Shows the conjugate multiplier: Displays the exact conjugate used, so you can copy it onto paper.
• • Surfaces the new denominator value: Returns the integer new denominator separately, so you can verify the size of the final fraction.
• • Reduces the final fraction: Divides numerator and new denominator by their GCD, keeping the result in lowest terms.
• • Handles negative new denominators: Multiplies both by -1 when the new denominator is negative, so the result is always written with a positive denominator.
• • Displays the intermediate step: Shows the fraction after the conjugate is distributed but before the final simplification.

The hardest part of rationalizing by hand is keeping the signs and the new denominator straight. The calculator names every quantity in the formula, so paper work can be checked against the panel row by row.

For the decimal value of the rationalized form, the Fraction to Decimal Calculator reads off the equivalent decimal in one step.

Three factors decide what the rationalized form looks like, and two limitations are worth knowing.

• • Cube roots and higher-order radicals are not handled; the formula is built for square-root denominators only.
• • The symbolic result keeps radicands in integer form and does not factor out perfect squares such as rewriting sqrt(8) as 2*sqrt(2).

According to Math Is Fun, rationalizing the denominator means removing radicals from the bottom of a fraction by multiplying by a clever form of 1 that keeps the value the same, with the standard trick being to multiply by the conjugate when the denominator is a sum or difference.

According to Wikipedia, rationalising a denominator removes radicals from the bottom of a fraction by multiplying numerator and denominator by a clever form of 1, with the conjugate a*sqrt(b) -/+ c*sqrt(d) used for the binomial case and the new denominator becoming a^2*b - c^2*d.

To combine several rationalized fractions over a common denominator, the Adding Fractions Calculator takes over.