How this calculator works
A radical is in simplest form when no square factor is left inside the root. To simplify √72, look for the largest perfect square dividing 72: 72 = 36 × 2, so √72 = 6√2. Adding radicals works like adding like terms and needs the same radicand: 2√2 + 3√2 = 5√2. Multiplying uses √a × √b = √(ab), and dividing uses √a ÷ √b = √(a/b). Each result is simplified again.
Formula
√(a²b) = a√b. √a × √b = √(ab). √a ÷ √b = √(a/b) for b > 0. Two radicals add only when their simplified radicands match: k√r + m√r = (k + m)√r.
Worked example
√72 = 6√2. Adding, √8 + √18 = 2√2 + 3√2 = 5√2. Multiplying, √2 × √8 = √16 = 4. Dividing, √72 ÷ √8 = √9 = 3.
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Frequently asked questions
Can I add √2 and √3 to get √5?
No. Radicals add like terms only when the radicands match. √2 + √3 is already in simplest form and is about 3.146, while √5 is about 2.236.
Why is √a + √b not √(a+b)?
The square root does not distribute over addition. A quick check with a = 9 and b = 16 shows it: √9 + √16 = 7, while √(9 + 16) = 5.
How do I know a radical is fully simplified?
Factor the radicand and check that no prime appears twice. If every prime exponent is 1, there is no square factor left to pull out, and the radical is simplified.
What is the decimal value for?
It is a check and a sense of size. The exact radical form is the primary answer; the decimal is rounded and cannot show whether the result is rational.