Rationalize radical denominators using conjugates and exact algebra. Get simplified expressions, decimal checks, and exportable results. Practice surd division steps with confidence and better algebra accuracy.
Use the form for expressions in this format: n / (b + c√d)
| Input | Conjugate | Rationalized Form | Approximate Value |
|---|---|---|---|
| 1 / (3 + √2) | 3 - √2 | (3 - √2) / 7 | 0.2265409197 |
| 2 / (5 + 2√3) | 5 - 2√3 | 2(5 - 2√3) / 13 | 0.1188021535 |
| 4 / (7 + √5) | 7 - √5 | 2(7 - √5) / 11 | 0.4335585997 |
For an expression written as n / (b + c√d), multiply the numerator and denominator by the conjugate (b - c√d).
Step 1: n / (b + c√d) × (b - c√d) / (b - c√d)
Step 2: Numerator becomes n(b - c√d)
Step 3: Denominator becomes b² - c²d
Final Form: n(b - c√d) / (b² - c²d)
This removes the radical from the denominator and gives an exact rationalized form.
Enter the numerator in the first field.
Enter the rational part of the denominator in the second field.
Enter the coefficient of the square root in the third field.
Enter the radicand in the final field.
Click calculate to view the conjugate, expanded terms, rationalized answer, and decimal check.
Use the export buttons to save the result as CSV or PDF.
A denominator rationalize calculator helps students rewrite fractions that contain square roots in the denominator. This algebra skill appears in school mathematics, competitive tests, and problem solving practice. A rationalized answer is easier to read, compare, simplify, and check. It also follows common formatting rules used in textbooks and exams.
This tool focuses on expressions shaped like n / (b + c√d). It uses the conjugate method. The conjugate changes the sign between the rational term and the radical term. When both parts are multiplied, the middle radical terms cancel. That leaves a rational denominator. The calculator then shows the expression, conjugate, expanded numerator, expanded denominator, and final rationalized form.
This page is useful for homework checking, guided practice, and concept review. Instead of showing only one final answer, it displays the main algebra steps. That makes it easier to verify manual work and understand why the method works. The decimal check adds another layer of confidence because it confirms the exact result numerically.
Use this calculator when you need fast algebra support without losing clarity. It works well for surd fractions, denominator cleanup, and exact expression review. The example table provides common forms, while the formula section explains the underlying rule. The export tools also help students and teachers save results for notes, worksheets, or revision files.
The layout stays simple and clean. The result appears above the form after submission, which makes review easier. The calculator fields stay responsive across large screens, tablets, and mobile devices. This creates a practical learning page for anyone studying rationalization of denominators in algebra.
It means removing a radical from the denominator of a fraction. The expression keeps the same value, but the final form becomes easier to read, simplify, and compare in algebra work.
The conjugate changes the sign between denominator terms. Multiplying by it creates a difference of squares. That removes the square root from the denominator and leaves a rational value below.
This calculator supports fractions written as n / (b + c√d). You can also use a negative coefficient for c to represent subtraction inside the denominator.
Yes. The numerator, rational term, and root coefficient accept decimals. The radicand should stay positive, and whole numbers work best for standard surd expressions.
The calculator shows an error message. This prevents undefined results and tells you to change the entered values before calculating again.
Yes. It reduces common whole-number factors where possible and also simplifies a single square root term when the radicand contains a perfect-square factor.
The decimal value helps verify the exact answer. You can compare the original expression and the rationalized form numerically to confirm both represent the same quantity.
Students, teachers, tutors, and exam candidates can all use it. It is helpful for algebra lessons, homework review, worksheet creation, and quick denominator rationalization practice.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.