Rationalize Denominator Calculator - Radical Fractions
Rationalize fractions with radical denominators in simple or binomial form and see the conjugate method spelled out step by step.
Choose a denominator type, enter the numeric values, and convert a radical denominator into an equivalent fraction with a rational denominator.
Rationalize Denominator Calculator - Radical Fractions
Rationalize fractions with radical denominators in simple or binomial form and see the conjugate method spelled out step by step.
About the rationalize denominator calculator
Rationalizing a denominator means rewriting a fraction so that no radical remains in the denominator. The value of the fraction does not change. You simply multiply the numerator and denominator by a carefully chosen expression equal to 1. In introductory algebra, the most common target is a denominator containing a square root, because expressions such as 3/√5 or 2/(3 + √2) are easier to compare, simplify, and use in later formulas after the radical is moved to the numerator.
For a simple radical denominator like a/√b, the idea is direct: multiply by √b/√b. The denominator becomes √b × √b = b, which is rational, while the numerator becomes a√b. The result is (a√b)/b. This is the pattern many students first learn when simplifying surds, and it shows up in geometry, trigonometry, and physics whenever exact radical forms appear.
A binomial denominator such as c + √b or c - √b requires the conjugate. The conjugate changes the sign between the two terms: the conjugate of c + √b is c - √b, and the conjugate of c - √b is c + √b. When you multiply a binomial by its conjugate, the middle radical terms cancel and you get a difference of squares: (c + √b)(c - √b) = c² - b. That cancellation is the key reason conjugates are so useful. It replaces a messy radical denominator with a clean rational number.
This rationalize denominator calculator focuses on the two algebra patterns that cover most classroom problems. In simple mode, you enter the numerator and the radicand and the tool returns the rationalized fraction and decimal value. In binomial mode, you enter the numerator, the rational part c, the sign, and the radical part b. The calculator shows the conjugate, the denominator simplification, the final rationalized expression, and a decimal check so you can confirm the equivalent value.
Understanding the method is more important than memorizing the finished form. Rationalization is not a trick for changing the answer; it is a technique for rewriting the same quantity in a more usable format. Whether you are simplifying an algebra homework problem, preparing an exact form for calculus, or checking symbolic manipulation by hand, this calculator helps you move from radical denominators to rational denominators without skipping the reasoning.
Rationalize denominator examples
These examples cover both the simple radical case and the binomial conjugate case.
| Input | Result | Explanation |
|---|---|---|
| Simple mode: a = 3, b = 5 | (3√5)/5 | Start with 3/√5 and multiply by √5/√5. The denominator becomes 5 and the numerator becomes 3√5. |
| Binomial mode: a = 2, c = 3, sign = +, b = 2 | 2(3 - √2)/7 | Start with 2/(3 + √2) and use the conjugate 3 - √2. The denominator becomes 3² - 2 = 7. |
| Binomial mode: a = 4, c = 5, sign = −, b = 6 | 4(5 + √6)/19 | Start with 4/(5 - √6) and multiply by the conjugate 5 + √6. The denominator simplifies to 25 - 6 = 19. |
How to use the rationalize denominator calculator
- Choose Simple (√b) for a denominator made of only one square root, or Binomial (c ± √b) when a rational term and radical are added or subtracted.
- Enter the numerator and the denominator values for the mode you selected. In binomial mode, also pick whether the denominator uses a plus or minus sign.
- Click Rationalize to see the conjugate or multiplier, the denominator simplification, and the final rationalized fraction.
- Use the decimal value to verify that the rationalized expression is equivalent to the original fraction.
Rationalize denominator FAQ
Why do mathematicians rationalize denominators?
A rational denominator is often easier to compare, simplify, and combine with other expressions. In many algebra and calculus settings, it is considered the standard exact form.
What is a conjugate?
For a binomial involving a radical, the conjugate keeps the same terms but changes the sign between them. The conjugate of c + √b is c - √b, and vice versa.
Does rationalizing change the value of the fraction?
No. You multiply the numerator and denominator by the same nonzero expression, which is equivalent to multiplying by 1. The expression looks different but represents the same number.
Why does the denominator become c² - b in binomial mode?
Because multiplying a binomial by its conjugate creates a difference of squares: (c + √b)(c - √b) = c² - (√b)² = c² - b.
Can I use negative or decimal numerators?
Yes. The calculator accepts any real numerator and real rational part c. The only restriction is that the value under the square root must stay positive and the denominator cannot evaluate to zero.