Square Root Calculator
The square root, cube root or any root of a number, with the simplified radical.
Good to know. The square root of 2 has been calculated to trillions of digits, and it still never repeats.
How it works
In plain words
The n-th root of a number is the value that, multiplied by itself n times, gives the number. To simplify, take out every factor that appears n times: 72 = 36 × 2, and √36 = 6, so √72 = 6√2.
The formula
ⁿ√x = x^(1/n) √(a² × b) = a√b
Worked example
√72 = 8.4852813742, because 8.4852813742 × 8.4852813742 = 72. As 72 = 36 × 2, it simplifies to 6√2.
Sources
No outside data or rates: this is plain arithmetic.
6 automated tests check its answers on every change.
About the square root calculator
The square root of a number is the value that, multiplied by itself, gives that number: √49 = 7 because 7 × 7 = 49. The cube root does the same with three factors (∛27 = 3), and the n-th root with n of them.
Most square roots are not whole numbers. They are irrational: the decimals go on forever without repeating, so the result is rounded to 10 decimal places. Numbers with a whole-number root, like 1, 4, 9, 16, 25 and 36, are called perfect squares; 8, 27 and 64 are perfect cubes. For other whole numbers, the calculator shows the two perfect powers either side.
In algebra and exams, roots are usually written in simplified radical form rather than as decimals. Split the number into a perfect square times what is left, and take the square root of the perfect square: √50 = √(25 × 2) = 5√2. The calculator does this for any whole number and any root.
Negative numbers have no real square root, since any number times itself is positive; in mathematics, √−1 is the imaginary number i. Odd roots of negative numbers do exist: ∛−8 = −2.
Questions people ask
What is the square root of 2?
About 1.4142135624. It is irrational, so the decimals never end or repeat.
How do I simplify a square root?
Find the largest perfect square that divides the number, and take its root outside: √72 = √(36 × 2) = 6√2.
How do I find a square root without a calculator?
Guess, divide the number by your guess, and average the guess and the result. Repeat: for √10, 3 → (3 + 3.33) ÷ 2 = 3.17 → 3.1623.
Can you take the square root of a negative number?
Not in real numbers. In complex numbers, √−9 = 3i. Cube roots of negative numbers are fine: ∛−27 = −3.