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How to divide radicals — step by step (with a worked example)

Q: What is the quotient rule for radicals?

The quotient rule is √a ÷ √b = √(a ÷ b), valid for a ≥ 0 and b > 0. It lets you combine two separate roots into a single root of the quotient.

Q: How do I compute √12 ÷ √3?

Apply the quotient rule: √12 ÷ √3 = √(12 ÷ 3) = √4. Because 4 is a perfect square, √4 = 2. The result is exactly 2.

Q: How do I rationalize the denominator?

Multiply the fraction by the root from the denominator: √a/√b = (√a · √b)/(√b · √b) = √(a · b)/b. So √2/√3 becomes √6/3. Since √b · √b = b, the root in the denominator disappears.

Q: How do I simplify √6 ÷ √2?

√6 ÷ √2 = √(6 ÷ 2) = √3 ≈ 1.7321. Because 3 has no perfect-square factor, √3 is already in its simplest form.

Q: Why shouldn't a root be left in the denominator?

A rational (root-free) denominator is the usual standard form. Such fractions are easier to compare, add, and work with further, which is why we rationalize the denominator.

Q: Can the numbers under the root be negative?

Not for real square roots: the radicand must be ≥ 0, and when dividing the denominator's radicand must even be > 0, because you can't divide by 0.

To divide two square roots, use the quotient rule √a ÷ √b = √(a ÷ b): divide the numbers under the roots (the radicands) and take a single root. Worked example: √12 ÷ √3 = √4 = 2. If a root is left in the denominator, you rationalize it. This is part of Grade 9 radical arithmetic.

By Valerij Hofmann · Fullstack Developer

Updated May 29, 2026Intermediate

Quick answer

To divide two square roots, use the quotient rule: √a ÷ √b = √(a ÷ b). You divide the radicands and take a single root. Example: √12 ÷ √3 = √(12 ÷ 3) = √4 = 2. If the quotient isn't a perfect square, rationalize the denominator — so √2/√3 becomes √6/3.

At a glance

Summary of this tutorial
Problem	√12 ÷ √3
Method	Quotient rule √a ÷ √b = √(a ÷ b)
Steps	Divide radicands, then take the root
Result	√4 = 2
Special case	Root in denominator → rationalize (√2/√3 = √6/3)
Grade level	Grade 9 (ages 14–15)

Worked example: √12 ÷ √3

EXAMPLE

√12 ÷ √3

We divide the radicands 12 ÷ 3 = 4 and take the root: √4 = 2.

The steps to divide two square roots

These steps work for any division of the form √a ÷ √b with a ≥ 0 and b > 0.

Step 1 · Start
√12 ÷ √3
Two square roots are to be divided.
Step 2 · Quotient rule
√(12 ÷ 3)
Combine √a ÷ √b into √(a ÷ b).
Step 3 · 12 ÷ 3
√4
Work out the quotient under the root.
Step 4 · Root
= 2
4 is a perfect square, so √4 = 2 exactly.

Why the quotient rule works

A square root is the inverse of squaring, and powers distribute over division: (a/b) to the power ½ equals a^½ ÷ b^½. That is exactly what √a ÷ √b = √(a ÷ b) says. Intuitively, instead of taking two roots separately, you first divide the radicands and take a single root — the result is the same. When a root stays in the denominator, you multiply by it because √b · √b = b makes the denominator root-free without changing the value.

Practice it yourself

Open the radical calculator

Enter your own square roots. See every step.

Start practice set

Questions on dividing radicals

Frequently asked questions

How do you divide two square roots?

Use the quotient rule: √a ÷ √b = √(a ÷ b). Divide the numbers under the roots and take a single root. Example: √12 ÷ √3 = √(12 ÷ 3) = √4 = 2. If the quotient isn't a perfect square, simplify the root or rationalize the denominator.

What is the quotient rule for radicals?

How do I compute √12 ÷ √3?

How do I rationalize the denominator?

How do I simplify √6 ÷ √2?

Why shouldn't a root be left in the denominator?

Can the numbers under the root be negative?

Quick answer

At a glance

Worked example: √12 ÷ √3

The steps to divide two square roots

Step 1 · Start

Step 2 · Quotient rule

Step 3 · 12 ÷ 3

Step 4 · Root

Why the quotient rule works

Practice it yourself

Frequently asked questions