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Square Root Simplifier

Simplify any square root into coefficient times a smaller radical.

Result

6√2
8.485

How it is calculated

The input must be a whole number from 0 up to 9,007,199,254,740,991; a decimal such as 72.4 or a negative number shows no result. The tool strips the square factors of each prime in turn: if a prime p divides n an even number of times e, pe/2 moves outside the root; if it divides an odd number of times, one copy stays inside. The product of what moved outside is the coefficient c, and √n = c × √r, where r is what remains under the radical. If r ends up as 1 only the whole number is shown (√49 gives 7); if c is 1 the root stays as √r. The decimal value is √n to three places. Zero returns 0.

Worked example

Simplify √72. Factor it: 72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3². The 3s form a pair, so one 3 moves outside. The 2s give one pair, so one 2 moves outside, with a single 2 left inside. Coefficient: 2 × 3 = 6, remainder 2. So √72 = √(36 × 2) = 6 × √2, shown as 6√2. The decimal value is 8.485, which matches 6 × 1.414.

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Common mistakes

  • Taking out a square that is not the largest. √72 = 2√18 is true but unfinished, since 18 still contains 9. The tool pulls out every square factor, giving 6√2.
  • Entering a decimal such as 72.4 and expecting a decimal root. Only whole numbers are accepted and a decimal gives no result.
  • Entering a negative number. There is no real square root, and the tool does not give an imaginary form such as i√8.
  • Dropping the coefficient. 6√2 means 6 times √2, not √(6 × 2); entering 12 gives 2√3, which is a different number.
  • Expecting a simplified form for every number. A number with no square factor, such as 17, stays √17, and only the decimal 4.123 helps there.

About this calculator

Simplifying a square root means rewriting it as a coefficient times the smallest possible radical, by pulling out any perfect-square factors: √72 becomes 6√2 because 72 = 36 × 2 and 36 is a perfect square. The value doesn’t change — 6√2 and √72 are the exact same number — but the simplified form is far easier to work with by hand.

Simplified radicals are the expected answer format in algebra and geometry classes, since unsimplified roots make it much harder to combine like terms (√8 + √18 doesn’t obviously simplify, but 2√2 + 3√2 = 5√2 does), rationalize denominators, or apply the quadratic formula cleanly when the discriminant isn’t a perfect square. The same factoring technique is also the basis for simplifying radical expressions in trigonometry (exact values like √3/2) and for reducing surds in physics formulas.

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