Square Root Calculator
Find the square root of any number, its simplest radical form such as 6√2, and cube or higher roots, with the prime factors shown.
A square root calculator finds the number that, multiplied by itself, gives the number you enter. The square root of 72 is 8.485281374, which simplifies to 6√2 because 72 is 36 times 2 and 36 is a perfect square. This calculator gives both the decimal and the simplest radical form for any whole number up to one trillion, says whether the number is a perfect square, and handles cube roots up to 10th roots.
This calculator does not handle complex roots of negative numbers or simplify roots of fractions and decimals; decimal results are rounded to 10 significant digits, and the radical forms are the exact answers.
√72
8.485281374
- Simplest radical form
- 6√2
- Perfect square
- No
- Prime factors
- 2^3 × 3^2
- Between the perfect squares
- 64 and 81
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About the Square Root Calculator
Simplifying a square root means pulling every perfect square out from under the radical sign. The calculator does it the way a textbook does, by prime factors: 72 is 2 × 2 × 2 × 3 × 3, each pair of equal primes comes out as a single factor, and whatever has no partner stays inside. Two 2s and two 3s pair off to give 2 × 3 = 6 outside, the leftover 2 stays inside, and the answer is 6√2. The same idea works for other roots with groups instead of pairs: a cube root needs three of a kind, so the cube root of 48, which is 2 × 2 × 2 × 2 × 3, becomes 2∛6. When nothing is left inside, the number was a perfect power, as with 144, whose square root is exactly 12. Negative numbers have real cube roots, so the cube root of -27 is -3, but no real square root, and the calculator says so instead of showing an error. The decimal square root is rounded the way computer arithmetic standards require, to the nearest value a double can hold, and shown to 10 significant digits.
Frequently asked questions
How do I simplify a square root by hand?+
Find the largest perfect square that divides the number, take its root outside, and leave the rest inside. For 50, the largest perfect square factor is 25, so √50 = √25 × √2 = 5√2, about 7.071067812. If you cannot spot the square, list the prime factors and pair them up, as the calculator shows under prime factors.
What is the square root of 2?+
About 1.414213562. It cannot be simplified, because 2 has no square factor other than 1, and it never ends or repeats as a decimal, so any decimal you see is rounded. That is why exact answers in algebra are left as √2.
How can I estimate a square root without a calculator?+
Find the two perfect squares on either side. The square root of 72 lies between 8 and 9, because 72 is between 64 and 81, and it sits a little closer to 8 since 72 is closer to 64. The calculator lists those neighbouring perfect squares under every result that is not itself a perfect square.
Can it do cube roots and other roots?+
Yes. Choose cube root or any root up to the 10th from the menu. Odd roots of negative numbers work, so the cube root of -27 is -3; even roots of negative numbers have no real answer. Whole numbers up to one trillion also get their simplest radical form, such as 2∛6 for the cube root of 48.
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