Dividing Radicals Calculator (simplify & rationalize the denominator)

Understanding the dividing radicals calculator — concept and principle

This calculator divides one radical by another, a·ⁿ√x ÷ c·ⁿ√y, and simplifies the result: it reduces the fraction under the radical, rationalizes the denominator and takes out perfect powers. It shows each step and the decimal value. Use it for algebra homework on square roots and cube roots.

Divide outside numbers and inside numbers separately, then clear the root from the bottom.

How it is calculated

Divide the coefficients and combine the radicals: ⁿ√x ÷ ⁿ√y = ⁿ√(x/y); reduce x/y to p/q.
Rationalize: multiply top and bottom by ⁿ√(qⁿ⁻¹), so the denominator becomes ⁿ√(qⁿ) = q.
Take every perfect n-th power out of the radicand and reduce the outside fraction.

Whole numbers only; radicands must be positive.

The principle

Roots respect multiplication and division: √(a/b) = √a ÷ √b. That lets two radicals merge into one, where the fraction can be reduced. By convention a simplified answer has no radical in the denominator. Multiplying by a carefully chosen form of 1, such as √3/√3, turns the denominator into a whole number without changing the value. Taking out perfect squares (or cubes) finishes the simplification, for example √12 = 2√3.

Worked example

Default inputs: square root (n = 2), numerator 6√10, denominator 3√15.

Problem: 6√10 ÷ 3√15.
Step 1: 6 ÷ 3 = 2 and √(10/15) = √(2/3), giving 2·√(2/3).
Step 2: multiply top and bottom by √3: √2 ÷ √3 = √6 ÷ 3.
Step 3: no perfect squares remain in 6, so √6 / 3, and the result is 2√6/3.
Decimal value = 2 × 2.4495 ÷ 3 = 1.632993162.

How to read the result

The simplified form is exact; the decimal is for estimating. Both 2√(2/3) and 2√6/3 have the same value, but the second is the standard final form because its denominator is rational. Check by squaring: (2√6/3)² = 24/9 = 8/3, and (6√10 ÷ 3√15)² = 360/135 = 8/3.

Common mistakes

Dividing radicands that are under different indices, such as a square root by a cube root.
Rationalizing with the whole denominator 3√15 instead of just the radical part.
Forgetting to reduce x/y before rationalizing, which makes the numbers larger.
Leaving a perfect square inside the radical.

Keep in mind: The rules used here assume positive radicands; with negative numbers and even roots, real-number rules break down.

Frequently asked questions

How do you divide square roots?

Divide the numbers under one root, √a ÷ √b = √(a/b), then simplify.

What does rationalizing the denominator mean?

Rewriting a fraction so no radical is left in the denominator.

What is 6√10 ÷ 3√15?

2√6/3, about 1.633.

Sources: OpenStax, Intermediate Algebra 2e, §8.5 Divide Radical Expressions

Checked 9 Nov 2026. OpenStax, Intermediate Algebra 2e (2020), §8.5 Divide Radical Expressions (quotient property ⁿ√(a/b) = ⁿ√a ÷ ⁿ√b; rationalize a one-term denominator by multiplying by the radical that makes the radicand a perfect power of the index).

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