Simplify Square Root Calculator: simplest radical form with steps
Reduce a square root to simplest radical form, e.g. √72 = 6√2, with the prime factorization, the perfect-square factor pulled out, the decimal value and step-by-step working. Also handles a coefficient, decimals and fractions, negative numbers (i) and cube or higher roots.
How it works
√n = √(k² × m) = k√m, where k² is the largest perfect-square factor of n · a√n = (a × k)√m · ⁿ√x: take out every group of n equal prime factors · √(p/q) = √(p × q) ÷ q · √(−n) = i√n
Simplifying a square root means pulling the largest perfect-square factor outside the radical. The calculator finds the prime factorization of the number, takes each pair of equal primes outside the root as a single factor, and leaves the unpaired primes inside. What remains inside has no square factor other than 1, so the root is in simplest radical form. A perfect square simplifies to a whole number with nothing left under the root. Under More options you can multiply by a coefficient (for 3√72 the result is 3 × 6√2 = 18√2) and switch to a cube, fourth, fifth or sixth root, where groups of 3, 4, 5 or 6 equal primes come out instead of pairs. Decimals are written as fractions and the denominator is rationalized, so √0.5 = √2/2. A negative number under an even root gives an imaginary result with i = √−1, while an odd root of a negative number is simply negative. Numbers up to one trillion are factored exactly.
Worked example
For √72: the prime factorization is 72 = 2³ × 3². One pair of 2s and the pair of 3s come out as 2 × 3 = 6, and one 2 stays inside. So the largest perfect-square factor is 36 = 6², 72 = 36 × 2, and √72 = √36 × √2 = 6√2 ≈ 8.485281.
Frequently asked questions
What is simplest radical form?
A square root is in simplest radical form when the number under the root has no perfect-square factor other than 1, there is no fraction under the root and no root in a denominator. For example √8 becomes 2√2, and √2 is already simplest.
How do you simplify a square root step by step?
Find the largest perfect square that divides the number (or write its prime factorization), split the root into √(perfect square) × √(rest), and take the square root of the perfect square. √12 = √4 × √3 = 2√3, √18 = √9 × √2 = 3√2 and √45 = √9 × √5 = 3√5.
What happens with a perfect square?
It simplifies completely to a whole number with no radical left: √49 = 7 and √100 = 10.
Why can’t I simplify √2 or √3?
Their radicands are prime, so they contain no perfect-square factor to pull out. They are irrational and already in simplest form. The same is true for any square-free number such as √6, √10 or √30.
Can you simplify the square root of a negative number?
Not as a real number, but in complex numbers √−n = i√n, so √−72 = 6i√2. A cube root of a negative number is real: ∛−54 = −3∛2.
How do you simplify the square root of a decimal or fraction?
Write it as a fraction and rationalize the denominator. 0.5 = 1/2, so √(1/2) = √2 ÷ 2 ≈ 0.707107. √0.12 = √(3/25) = √3/5.
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