Enter any non-negative number to simplify its square root to a√b form
√= ?
Result
√ =
Step-by-Step Derivation
Radical Simplification Formula
√N = a√b, where a²×b = N, and b has no perfect square factors
Extract perfect square factors (such as 4, 9, 16, 25...) from under the square root. The remaining part should contain no perfect square factors, giving the simplest radical form.
⚠Note: The radicand must be non-negative. If it is already a perfect square (e.g., 16), the result is an integer (e.g., 4). Perfect squares need no simplification.
What Is Radical Simplification?
Radical simplification is the process of factoring the radicand and extracting all perfect square factors out of the square root, resulting in the simplest radical form a√b.
Perfect Squares
Perfect squares are numbers that can be written as an integer squared: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100...
Prime Factorization
Factor the radicand into primes. Every pair of identical factors can be taken out of the square root once. E.g., 72 = 2³×3² → √72 = √(2²×3²×2) = 6√2.
Perfect Square → Integer
When the radicand itself is a perfect square, the square root can be removed entirely. For example, √36 = 6.
Why Simplify?
Simplified radicals are easier to compare and compute with. For instance, comparing √50 with 7 is easier when you know 5√2 ≈ 7.07.
A radical is in simplest form when the radicand contains no perfect square factors. For example, √18 = 3√2 is simplest form because 2 is not a perfect square.
Why does √18 equal 3√2?▼
Because 18 = 9 × 2, √18 = √(9×2) = √9 × √2 = 3√2. Since 9 is a perfect square, it can be taken out of the square root, while 2 cannot be factored as a product of perfect squares, making it the simplest radical form.
In the real number system, square roots of negative numbers are undefined. To handle them, we introduce complex numbers: the imaginary unit i = √(-1), so √(-18) = 3√2·i.
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