Last updated October 8, 2026 · Report an error
Dividing two square roots combines cleanly into a single radical, using the quotient rule for radicals: √a ÷ √b = √(a/b). Once combined, the resulting fraction under the root can often be simplified further, and if a radical still remains in the denominator after simplifying, the standard convention is to "rationalize" it — rewrite the expression so no irrational number sits in the denominator. This is a core skill in algebra coursework, and it also matters practically whenever a formula (in physics, engineering, or finance) produces a radical-over-radical expression that needs to be reported in its simplest, most standard form.
How it works
Enter the two radicands (the numbers under each square root). The calculator combines them into a single fraction under one radical using √a/√b = √(a/b), simplifies that fraction to lowest terms if possible, and then simplifies the resulting root by pulling out any perfect-square factors. If a radical remains in the denominator after all that simplification (which happens whenever the simplified fraction isn't itself a perfect square), the calculator rationalizes it by multiplying both the numerator and denominator by that same remaining radical — since any nonzero radical divided by itself equals exactly 1, this doesn't change the expression's value, but it does clear the radical out of the denominator.
- Enter numerator radicand (a) under √.
- Enter denominator radicand (b) under √.
- Click Calculate to see your results.
Examples
√8 ÷ √2
Combine: √(8/2) = √4 = 2 — a whole number, no rationalization needed.
√72 ÷ √8
Combine: √(72/8) = √9 = 3.
√3 ÷ √2
Combine: √(3/2). Since 3/2 isn't a perfect square, rationalize by multiplying top and bottom by √2: (√3×√2)/(√2×√2) = √6/2 ≈ 1.2247.
Who should use it
- Simplifying a radical expression for an algebra assignment.
- Cleaning up a formula result that contains a radical in the denominator.
- Checking a hand-simplified radical division for errors.
Industry applications
- Algebra education
- Physics and engineering formulas involving radical expressions
Advantages
- Combines, simplifies, and rationalizes in one consistent process.
- Produces the conventional, fully-simplified form expected in algebra coursework.
Limitations
- This tool specifically handles square roots — a different rationalizing factor is needed for cube roots or higher-index radicals.
Common mistakes to avoid
- Simplifying the numerator and denominator radicals separately but forgetting to rationalize if the denominator still has a radical left over.
- Combining √a ÷ √b into √(a÷b) but forgetting to simplify the resulting fraction to lowest terms before taking the root.
- Multiplying only the denominator by the rationalizing radical and forgetting to multiply the numerator by the same factor, which changes the expression's value.
Best practices
- Always combine the two radicals into one fraction under a single root first, then simplify that fraction, before worrying about rationalization.
- Check whether the simplified radicand is a perfect square before assuming rationalization is needed — sometimes the radical vanishes entirely.
- When rationalizing, multiply BOTH the numerator and denominator by the same radical to preserve the expression's value.
Tips
- Working with fractional exponents instead of radical notation? Use the Fraction Exponent Calculator to convert between the two forms.