Rationalize Denominator
Eliminate radicals from any fraction a/sqrtb with this free calculator. Shows the rationalized expression, simplified radical, and decimal value for algebra work.
About This Calculator
The Rationalize Denominator Calculator helps you eliminate square roots from the denominator of any fraction of the form a/sqrtb. This process, called rationalizing the denominator, is a fundamental algebra skill that produces a mathematically equivalent expression with a rational denominator. Students from middle school through college use this technique when simplifying radical expressions, solving equations, and preparing final answers in standard form.
The calculator applies the standard rationalization formula: multiply numerator and denominator by sqrtb to get (a x sqrtb) / b. For example, entering a = 5 and b = 2 gives the rationalized form (5sqrt2)/2, which equals approximately 3.5355. The tool automatically simplifies the square root when possible, extracting perfect square factors from the radicand for the cleanest result.
This calculator handles positive radicands only, as square roots of negative numbers produce imaginary results. It also detects perfect square radicands and reports them without the radical symbol since they evaluate to whole numbers. Whether you are checking homework, preparing for an exam, or teaching rationalization concepts, this tool provides both the exact symbolic result and the decimal approximation.
How to Use
Enter the numerator value (a) and the radicand (b, the number inside the square root) into the input fields. Click Calculate to see the rationalized form displayed as an exact expression, the original a/sqrtb form for reference, and the decimal approximation rounded to six decimal places.
Frequently Asked Questions
What does it mean to rationalize a denominator?
Rationalizing the denominator means rewriting a fraction with a radical (square root) in the denominator without changing its value. The goal is to eliminate the radical from the denominator by multiplying both numerator and denominator by the radical. For example, 5/sqrt2 becomes (5sqrt2)/2, which has the same value but no radical in the denominator.
How do you rationalize a denominator with a single square root?
For a fraction of the form a/sqrtb, multiply both numerator and denominator by sqrtb. This gives (a x sqrtb) / (sqrtb x sqrtb) = (asqrtb) / b. Since sqrtb x sqrtb equals b, the denominator becomes a rational number with no radical.
Why do we rationalize denominators in math?
Rationalizing denominators is a standard convention in mathematics because it simplifies calculations, makes it easier to compare fractions, and is required for standard form in algebra textbooks. Having a radical in the denominator is considered less simplified and harder to work with in further computations.
Can you rationalize denominators with cube roots?
Yes, the same principle applies. For a cube root in the denominator, multiply numerator and denominator by the cube root squared to eliminate it. For example, for a/∛b, multiply by (∛b)^2/(∛b)^2 to get (a∛b^2)/b. The denominator becomes rational because ∛b x ∛b x ∛b = b.
What is the rationalized form of 3/sqrt5?
The rationalized form of 3/sqrt5 is (3sqrt5)/5. Multiply numerator and denominator by sqrt5: (3 x sqrt5)/(sqrt5 x sqrt5) = (3sqrt5)/5. The decimal value is approximately 1.3416.
How do you simplify when the radicand is a perfect square?
If the radicand b is a perfect square, then sqrtb is an integer, so no rationalization is needed. For example, 7/sqrt9 = 7/3 directly because sqrt9 = 3. The calculator automatically detects this case.
Is this calculator free to use?
Yes, this rationalize denominator calculator is completely free to use with no limits or registration required. Bookmark it for quick access during homework, exam prep, or any algebra work.