Dividing Radicals Calculator
Divide two square-root expressions of the form (c₁√a) ÷ (c₂√b). Enter the coefficients and radicands to get the quotient, the rationalized form with no radical in the denominator, and a worked step-by-step breakdown.
(2√18) ÷ (3√2) = 2
Write the expression
Separate coefficients from radicals
Quotient rule: √a ÷ √b = √(a ÷ b)
Rationalized form — multiply by √b/√b
Decimal result
- 1
Divide coefficients
2 ÷ 3 = 0.6667 - 2
Apply quotient rule √a ÷ √b = √(a÷b)
√(18 ÷ 2) = √9 = 3 - 3
Multiply coefficient ratio by √(a÷b)
0.6667 × 3 = 2
How does this calculator work?
To divide (c₁√a) by (c₂√b): divide the coefficients to get c₁/c₂, apply the quotient rule √a÷√b = √(a/b), then rationalize by multiplying by √b/√b to get c₁√(ab)/(c₂b). Enter your coefficients and radicands for the decimal result, rationalized form, and step-by-step working.
Formula
How this is calculated
The quotient property of square roots states that √a ÷ √b = √(a/b) for a ≥ 0 and b > 0. To divide (c₁√a) by (c₂√b), first divide the coefficients (c₁/c₂) and then apply the quotient property to the radical parts, giving (c₁/c₂) × √(a/b). The result can then be evaluated numerically.
Rationalizing the denominator removes the square root from the denominator — the conventional form in algebra. Multiplying numerator and denominator by √b transforms (c₁√a)/(c₂√b) into c₁√(ab)/(c₂b). Because both parts of the fraction are multiplied by the same value, the numerical result is unchanged; only the form differs.
This calculator handles square roots only. Both radicands must be real and non-negative, the denominator radicand must be strictly positive, and the denominator coefficient must be non-zero to avoid division by zero. Floating-point arithmetic is used for the decimal result; expressions that yield exact integers or simple fractions are shown rounded to six decimal places.
Frequently asked questions
√a ÷ √b = √(a/b) for a ≥ 0 and b > 0. It works in both directions: you can split a radical over a fraction (√(9/4) = √9/√4 = 3/2) or combine two radicals into one (√8/√2 = √4 = 2).
Convention in algebra requires expressions to have no radical in the denominator. Multiplying top and bottom by √b converts √b in the denominator to b (rational), giving the canonical form. The value does not change — only the presentation.
Yes — any finite non-negative radicand and any non-zero coefficient (including decimals) are accepted. The formula (c₁/c₂) × √(a/b) is valid for all real inputs satisfying the constraints.
Also known as
TG we-Calculate Editorial Team. (2026). Dividing Radicals Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/dividing-radicals-calculator
TG we-Calculate Editorial Team. "Dividing Radicals Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/dividing-radicals-calculator.
TG we-Calculate Editorial Team, "Dividing Radicals Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/dividing-radicals-calculator
@misc{wecalculate_dividing_radicals_calculator, title = {Dividing Radicals Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/dividing-radicals-calculator}}, year = {2026}, note = {TG we-Calculate} }
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