Rationalize Denominator Calculator
Rewrite a fraction of the form n ÷ (a + b√r) with a rational denominator, then inspect the conjugate, exact result, decimal check, and algebraic steps.
Expression inputs
Required integer from – 1,000,000 to 1,000,000.
The non-radical part of the denominator.
Required integer from – 10,000 to 10,000.
Positive integer under the square-root sign, 1 to 1,000,000.
Rationalized result
Exact rationalized form
(15 – 5√2) / 7
Equivalent to the original expression; the denominator contains no radical.
Conjugate or multiplier
3 – √2
Difference of squares
7
Decimal value
1.1327045983
Simplified original
5 / (3 + √2)
The startup example is valid and its Excel workbook is ready.
Rationalized form: (15 – 5√2) / 7.
Step-by-step transformation
| Step | Expression | Why it works |
|---|---|---|
| 1 | 5 / (3 + √2) | The radicand is already square-free. |
| 2 | Multiply by (3 – √2) / (3 – √2) | This factor equals 1, so the expression keeps the same value. |
| 3 | 5(3 – √2) / (3² – (√2)²) | Conjugates cancel the middle radical terms. |
| 4 | (15 – 5√2) / 7 | Expand, simplify, normalize the sign, and reduce common integer factors. |
The table and downloaded workbook are generated from the same canonical calculation model, so changed inputs update both consistently.
How to use the rationalize denominator calculator
What this calculator does
This calculator rewrites a real-number expression in the form n / (a + b√r) so that its denominator is rational. It first simplifies the square root, determines whether the denominator is already rational, and otherwise multiplies by the appropriate conjugate. The result is an exact algebraic identity, not a numerical approximation. The decimal value is included only as a cross-check. The tool handles integer coefficients and a positive integer radicand; it does not parse arbitrary typed formulas, variables, cube roots, complex numbers, or nested radicals.
When to use it
Use it to check algebra homework involving a one-term or two-term square-root denominator, to verify that a conjugate was chosen with the correct sign, to simplify a radical before rationalizing, or to compare an exact radical answer with its decimal value. The underlying rules are covered in OpenStax's section on dividing radical expressions and rationalizing denominators.
How to calculate
- The calculator opens with the complete example 5 / (3 + √2), and a validated example XLSX is immediately available. Replace any sample value with your own integer.
- Enter Numerator n, then describe the denominator using Rational term a, Radical coefficient b, and Radicand r. A negative coefficient represents subtraction; for example, b = – 2 produces a – 2√r.
- Read Exact rationalized form first. Use Conjugate or multiplier, Difference of squares, and the transformation table to audit each algebraic step. Use Decimal value only to confirm equivalence.
- Select Download Excel to export the current validated inputs, outputs, and steps. Reset clears the demonstration and all calculated content; Download Excel then stays disabled until all four required fields again form a valid, nonzero denominator.
Input guide
Numerator n is a required whole integer from – 1,000,000 through 1,000,000; enter plain digits with an optional leading sign, such as 5 or – 12. Increasing it scales the complete fraction. Commas, decimals, scientific notation, and mixed text are rejected rather than silently reinterpreted. Rational term a is the required integer outside the radical in the denominator, such as 3 in 3 + √2. Changing a changes both the conjugate and the difference-of-squares denominator. Radical coefficient b is the required integer multiplying the square root, such as 1 in √2 or – 2 in – 2√7; zero is allowed and means the denominator has no radical term. Radicand r is the required positive integer from 1 through 1,000,000 under the square-root symbol. A value such as 12 is automatically simplified because √12 = 2√3. A perfect square such as 9 becomes rational, so the method may switch from “Conjugate” to “Already rational.”
Output guide
Exact rationalized form is the reduced exact answer with a positive integer denominator. Conjugate or multiplier shows the expression multiplied above and below; for a binomial denominator, the sign between terms is reversed. Difference of squares shows a² – b²r after the radical has been simplified; the final reduced denominator can be smaller when all coefficients share a common factor. Decimal value is a ten-decimal estimate driven by all four inputs. Simplified original shows the input after perfect-square factors have been removed from r. The summary pills repeat the selected method, the square-free radicand, and the final denominator. Each transformation-table row shows an exact expression and its reason; no chart is used because this calculation has no meaningful multi-point quantitative series.
Worked example
For the startup values n = 5, a = 3, b = 1, and r = 2, the expression is 5 / (3 + √2). The conjugate is 3 – √2. Multiplying numerator and denominator by that conjugate gives 5(3 – √2) / ((3 + √2)(3 – √2)). The denominator is 3² – (√2)² = 9 – 2 = 7, while the numerator expands to 15 – 5√2. Therefore the first-open result is (15 – 5√2) / 7, approximately 1.1327045983. OpenStax's overview of radicals and rational exponents provides broader context for simplifying roots before applying this step.
Why the conjugate removes the radical
A conjugate pair has identical terms with the middle sign reversed: a + b√r and a – b√r. Their product is a difference of squares, a² – (b√r)², so the cross terms cancel and the square root is squared away. Multiplying the original fraction by the conjugate divided by itself multiplies by 1, preserving the value while changing its form. This is why rationalization is an exact identity rather than an approximation.
A rational number can be written as a ratio of integers with a nonzero denominator, while square roots of non-perfect squares are irrational. OpenStax's explanation of rational and irrational real numbers clarifies that distinction. Rationalizing does not make the whole expression rational; it only removes the irrational radical from the denominator.