Rationalize denominators containing square roots, cube roots, or binomial radicals. See step-by-step conjugate multiplication, before/after comparison, and simplified results.
The Rationalize the Denominator Calculator eliminates irrational numbers from the denominator of a fraction in just one click, showing you every step of the process. Rationalizing denominators is a fundamental algebra skill required whenever you need to express a fraction in standard mathematical form — teachers, textbooks, and standardized tests all expect denominators to be free of radicals.
This calculator supports four common rationalization scenarios. For monomial radicals like a/√c you simply multiply by √c/√c. For binomial expressions like a/(b + √c) the tool uses the conjugate (b − √c) to create a difference of squares in the denominator. It also handles expressions with two different radicals a/(√c − √d) and cube roots a/∛c. Each mode walks you through the process step by step.
The visual before-and-after comparison makes it easy to see how the expression transforms, and the reference table shows all four techniques side by side so you can quickly identify which method to use for any problem. Eight presets cover the most common textbook problems, letting you practice instantly. Whether you are a student working through a homework set, a tutor illustrating the concept, or someone preparing for the SAT or ACT, this calculator makes rationalization fast, accurate, and transparent.
Rationalize the Denominator Calculator helps you solve rationalize the denominator problems quickly while keeping each step transparent. Instead of redoing long algebra by hand, you can enter Numerator (a), Denominator rational part (b), Radicand (c) once and immediately inspect Original Expression, Rationalized Form, Conjugate Used to validate your work.
This is useful for homework checks, classroom examples, and practical what-if analysis. You keep the conceptual understanding while reducing arithmetic mistakes in multi-step calculations.
Monomial: a/√c × (√c/√c) = a√c/c. Binomial: a/(b+√c) × (b−√c)/(b−√c) = a(b−√c)/(b²−c). Cube root: a/∛c × (∛c²/∛c²) = a·∛c²/c.
Result: Original Expression shown by the calculator
Using the preset "1/(√2)", the calculator evaluates the rationalize the denominator setup, applies the selected algebra rules, and reports Original Expression with supporting checks so you can verify each transformation.
This calculator takes Numerator (a), Denominator rational part (b), Radicand (c), Second radicand (d) and applies the relevant rationalize the denominator relationships from your chosen method. It returns both final and intermediate values so you can audit the process instead of treating it as a black box.
Start with the primary output, then use Original Expression, Rationalized Form, Conjugate Used, Decimal Value to confirm signs, magnitude, and internal consistency. If anything looks off, change one input and compare the updated outputs to isolate the issue quickly.
A strong workflow is manual solve first, calculator verify second. Repeating that loop improves speed and accuracy because you learn to spot common setup errors before they cost points on multi-step algebra problems.
Convention in mathematics requires denominators to be rational. It also makes expressions easier to compare, add, and simplify further.
The conjugate of (a + √b) is (a − √b). Multiplying an expression by its conjugate creates a difference of squares, eliminating the radical.
No — you multiply by a form of 1 (e.g., √c/√c = 1), so the value stays the same while the form changes.
Multiply numerator and denominator by ∛c², so the denominator becomes ∛c³ = c, a rational number. Use this as a practical reminder before finalizing the result.
Use the two-radicals mode. The conjugate of (√a − √b) is (√a + √b), giving denominator a − b.
Yes — for an nth root, multiply by the (n−1)th power of the radicand under the root. This calculator covers square and cube roots, the most common cases.