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Dividing Radicals Calculator

Divide square roots online — free and step by step. Simplify √a ÷ √b with the quotient rule and rationalize the denominator, with the full working.

Quick answer
How do you divide two square roots?
Use the quotient rule: √a ÷ √b = √(a ÷ b). Example: √12 ÷ √3 = √(12 ÷ 3) = √4 = 2. If the quotient isn't a perfect square, rationalize the denominator: √a/√b = √(a · b)/b. So √2/√3 becomes √6/3.
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Step-by-step
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HowTo

Dividing radicals — step by step

Using “√12 ÷ √3” and “√2 ÷ √3”
  1. 1
    Step 1 of 4

    Apply the quotient rule

    Divide two square roots by dividing the radicands: √a ÷ √b = √(a ÷ b). √12 ÷ √3 becomes √(12 ÷ 3) = √4.

  2. 2
    Step 2 of 4

    Work out the quotient

    Divide under the root: 12 ÷ 3 = 4. If it's a perfect square, take the root exactly: √4 = 2.

  3. 3
    Step 3 of 4

    Simplify or rationalize

    No clean root? Pull out perfect squares (√18 = 3√2) or rationalize the denominator: √2/√3 multiplied by √3 → √6/3.

  4. 4
    Step 4 of 4

    State the result

    Write the simplified form, optionally with a decimal: √2/√3 = √6/3 ≈ 0.8165.

Examples

Dividing radicals — worked examples

Exact values, simplification, and rationalization
√12 ÷ √3
√(12 ÷ 3)
√4
2
√50 ÷ √2
√(50 ÷ 2)
√25
5
√6 ÷ √2
√(6 ÷ 2)
√3
√3 ≈ 1.7321
√2 ÷ √3
multiply by √3
√6 ÷ 3
√6/3 ≈ 0.8165
√5 ÷ √2
multiply by √2
√10 ÷ 2
√10/2 ≈ 1.5811
√27 ÷ √3
√(27 ÷ 3)
√9
3
Theory

Dividing radicals — the quotient rule and rationalizing

Square roots obey the quotient rule: √a ÷ √b = √(a ÷ b), as long as a ≥ 0 and b > 0. Instead of taking two roots separately, you divide the radicands and take a single root. If the quotient is a perfect square (4, 9, 16, 25, …) the result is exactly a whole number: √12 ÷ √3 = √4 = 2. Otherwise you simplify the root by pulling out perfect squares — for example √18 = √(9 · 2) = 3√2. If a root remains in the denominator, it is standard to rationalize it, that is to make the denominator root-free. You do this by multiplying by the denominator's root: √a/√b = (√a · √b)/(√b · √b) = √(a · b)/b. So √2/√3 becomes √6/3. Then reduce the fraction if numerator and denominator share a factor. These rules belong to middle-school radical arithmetic and underpin simplifying expressions, solving equations with roots, and working with irrational numbers.

Pitfalls

Common mistakes when dividing radicals

Subtracting radicands instead of dividing

√12 ÷ √3 is √(12 ÷ 3) = √4, not √(12 − 3). The quotient rule divides the radicands.

Leaving a root in the denominator

1/√3 counts as unsimplified. Multiply by √3: √3/3. A rational denominator is the standard form.

Missing perfect squares

√18 isn't “done” — it's 3√2. Always check whether a square hides in the radicand.

Not reducing after rationalizing

√3 ÷ √12 gives √36/12 = 6/12 = 1/2. Reduce the fraction at the end.
FAQ

Frequently asked questions about dividing radicals

Glossary

Glossary — key terms explained simply

Radical (root)
The square root √a is the non-negative number whose square is a.
Radicand
The number under the root sign.
Quotient rule
√a ÷ √b = √(a ÷ b).
Rationalize
Make the denominator root-free by multiplying appropriately.
Perfect square
A square number like 4, 9, 16 whose root is a whole number.
Irrational number
A number like √2 that can't be written as a fraction.