Square Root Calculator with Simplified Radical
Enter a number from 0 to 1,000,000,000,000 to get its square root and radical form.
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Formula
The square root of n is the number r with r × r = n. The nearest perfect squares are floor(√n)² and ceil(√n)². To simplify, write n = k² × m with m as small as possible, giving √n = k√m.
Worked example
For 72: √72 = 8.485281 (to 6 places). 8² = 64 and 9² = 81, so 72 lies between them. Since 72 = 36 × 2, the simplified radical is 6√2.
Frequently asked questions
How do I find a square root by hand?
Find the perfect squares on either side of your number, then estimate between them. For 72, 64 (8²) and 81 (9²) bracket it, so the root is between 8 and 9. The exact value is 8.485281.
How do I simplify a square root?
Pull out the largest perfect square factor. 72 = 36 × 2, so √72 = √36 × √2 = 6√2. If no square factor greater than 1 exists, the radical is already simplified, as with √2.
What is a perfect square?
It is a whole number that is the square of another whole number, such as 1, 4, 9, 16 and 144 (12 × 12). Their square roots are whole numbers, so the two nearest squares are the number itself.
Can I take the square root of a negative number?
Not with real numbers, since no real number squared is negative. The root of a negative number is imaginary (√-4 = 2i), which this calculator does not handle.