Square Root Calculator

A square root of x is the number that, multiplied by itself, gives x — so √49 = 7 because 7 × 7 = 49. For a perfect square you can find it by factorising. For other numbers use an estimate and refine it: Newton's method takes the average of a guess g and x ÷ g, and repeats. The long-division method gives the digits one at a time without guessing.

Write the number as a product of primes, group the primes in pairs, and move one prime from each pair outside the root. For 72 = 2 × 2 × 2 × 3 × 3 there is a pair of 2s and a pair of 3s, so 2 × 3 = 6 comes out and one 2 stays inside: √72 = 6√2. If no prime repeats, as with √30, the root is already in simplest form.

The square root of 2 is about 1.41421356237, and its decimal digits go on forever without repeating, because √2 is irrational. It is the length of the diagonal of a square with sides of 1. This calculator gives it to as many as 50 decimal places: set Decimal places to 50 and enter 2.

Not among the real numbers, because any real number multiplied by itself is zero or positive. In the complex numbers the square root of −1 is written i, so √−16 = 4i and √−72 = 6√2 i. Odd roots are different: the cube root of −27 is the real number −3, because (−3) × (−3) × (−3) = −27.

A perfect square is a whole number that is the square of another whole number, such as 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100. Its square root has no decimal part. In prime factor form every prime appears an even number of times. The calculator shows a Perfect square badge when the root comes out exact.

Split the digits into pairs starting from the decimal point. Find the largest digit whose square fits the first pair, subtract, then bring down the next pair. Double the root found so far, and find the largest digit d where (20 × root + d) × d fits the new number. Repeat for each pair, adding 00 pairs for decimal places.

A square root of x is the number that gives x when two copies are multiplied (y × y), while a cube root gives x when three copies are multiplied (y × y × y). So √64 = 8 but ∛64 = 4. Cube roots exist for negative numbers in the real numbers, whereas square roots of negatives are imaginary.

The nth root of x is the number that gives x when raised to the power n, written ⁿ√x or x^(1/n). The square root is the 2nd root and the cube root is the 3rd. For example the 5th root of 1024 is 4, because 4⁵ = 1024. Choose Nth root and set Index n to any whole number from 2 to 100.

Only for numbers greater than 1. For numbers between 0 and 1 the square root is larger: √0.25 = 0.5 and √0.01 = 0.1. The roots of 0 and 1 are themselves. This is because multiplying two numbers smaller than 1 always gives a smaller result.

The decimal result is computed with exact big-integer arithmetic and rounded half up at the last place you choose, so all digits up to 50 decimal places are correct. The Newton's method table uses ordinary floating-point numbers to illustrate convergence and is good to about 15 significant digits. Simplified radical form is given for numbers up to 10¹² after scaling.