Calculator
Use integer coefficients. The numerator is m + n√s. The denominator is a ± b√r.
Visual Comparison
The graph varies the rational numerator part while keeping your radical settings fixed.
Example Data Table
| Numerator | Denominator | Rationalizing factor | Rationalized form |
|---|---|---|---|
| 1 | √2 | √2 | √2 / 2 |
| 5 | 3 + √7 | 3 - √7 | (15 - 5√7) / 2 |
| 2 + √3 | 4 - √2 | 4 + √2 | (8 + 2√2 + 4√3 + √6) / 14 |
| 3√5 | 2√10 | √10 | 3√2 / 4 |
Formula Used
- For a single radical denominator: k / (b√r) = k√r / (br).
- For a binomial denominator: 1 / (a + b√r) = (a - b√r) / (a² - b²r).
- For the opposite sign: 1 / (a - b√r) = (a + b√r) / (a² - b²r).
- For square factor simplification: √(q²r) = q√r.
- The calculator also expands (m + n√s)(a ∓ b√r).
How to Use This Calculator
- Enter the rational and radical parts of the numerator.
- Enter the rational part, radical coefficient, sign, and radicand of the denominator.
- Use zero for missing parts, such as no numerator radical.
- Select the decimal precision for the final approximation.
- Press the submit button to see the result above the form.
- Use the CSV or PDF button to save the calculation.
Understanding Rationalizing the Denominator
Rationalizing the denominator means removing a radical from the bottom of a fraction. The value of the fraction does not change. Only its form changes. This is useful because many algebra courses prefer exact answers with rational denominators. It also makes comparison easier in many problems.
Why the Method Works
The key idea is multiplying by one. When the denominator is √r, multiply the fraction by √r / √r. The denominator becomes r. When the denominator is a + b√r, multiply by its conjugate. The conjugate is a - b√r. Their product is a difference of squares.
Handling Binomial Denominators
A binomial denominator has two parts. One part may be rational. The other part contains a square root. The conjugate changes only the middle sign. This creates a clean denominator. The terms with radicals cancel in the denominator. The numerator still needs expansion and simplification.
Simplifying the Final Form
After multiplication, square factors should be pulled out of radicals. For example, √12 becomes 2√3. Then every term should be checked for a common factor. A factor can be reduced only when it divides the whole numerator and the whole denominator. Reducing one term alone is incorrect.
Study Benefits
This calculator shows each major step. It supports simple radicals, conjugates, radical numerators, decimal checks, and export options. Students can use it to verify homework. Teachers can use it to prepare examples. The graph gives another view of how the fraction changes when the numerator changes.
Common Mistakes
Many errors happen when the wrong conjugate is used. Other errors come from forgetting to multiply the numerator. Some students also reduce only one term in a sum. This tool helps avoid those mistakes by showing the denominator product, the expanded numerator, and the final simplified expression.
FAQs
What does rationalizing the denominator mean?
It means rewriting a fraction so the denominator has no radical. The value stays the same. The expression becomes easier to compare, simplify, and use in exact algebra answers.
When should I use a conjugate?
Use a conjugate when the denominator has two terms, such as a + b√r or a - b√r. Change only the sign between the two terms.
Can this calculator handle a radical numerator?
Yes. Enter the numerator as m + n√s. Use n as zero when the numerator has no radical part.
Why does the denominator become rational?
The conjugate creates a difference of squares. The radical terms cancel in the denominator, leaving a² - b²r, which is rational.
Can I use negative coefficients?
Yes. Negative integer coefficients are accepted. The calculator normalizes signs and reduces common factors when the final expression allows it.
Why is my result not a decimal only?
Exact radical form is usually preferred in algebra. A decimal is also shown, but it may be rounded and less exact.
What if the denominator product is zero?
Then the chosen values make rationalization invalid for that form. Change the rational part, radical coefficient, or radicand.
How do the export buttons work?
The CSV button saves table data. The PDF button creates a short report with the original expression, conjugate, final form, and decimal value.