How do I calculate square root in Python?

SymPy

Depending on your goal, it might be a good idea to delay the calculation of square roots for as long as possible. SymPy might help.

SymPy is a Python library for symbolic mathematics.

import sympy
sympy.sqrt(2)
# => sqrt(2)

This doesn't seem very useful at first.

But sympy can give more information than floats or Decimals:

sympy.sqrt(8) / sympy.sqrt(27)
# => 2*sqrt(6)/9

Also, no precision is lost. (√2)² is still an integer:

s = sympy.sqrt(2)
s**2
# => 2
type(s**2)
#=> <class 'sympy.core.numbers.Integer'>

In comparison, floats and Decimals would return a number which is very close to 2 but not equal to 2:

(2**0.5)**2
# => 2.0000000000000004

from decimal import Decimal
(Decimal('2')**Decimal('0.5'))**Decimal('2')
# => Decimal('1.999999999999999999999999999')

Sympy also understands more complex examples like the Gaussian integral:

from sympy import Symbol, integrate, pi, sqrt, exp, oo
x = Symbol('x')
integrate(exp(-x**2), (x, -oo, oo))
# => sqrt(pi)
integrate(exp(-x**2), (x, -oo, oo)) == sqrt(pi)
# => True

Finally, if a decimal representation is desired, it's possible to ask for more digits than will ever be needed:

sympy.N(sympy.sqrt(2), 1_000_000)
# => 1.4142135623730950488016...........2044193016904841204

Option 1: math.sqrt()

The math module from the standard library has a sqrt function to calculate the square root of a number. It takes any type that can be converted to float (which includes int) as an argument and returns a float.

>>> import math
>>> math.sqrt(9)
3.0

Option 2: Fractional exponent

The power operator (**) or the built-in pow() function can also be used to calculate a square root. Mathematically speaking, the square root of a equals a to the power of 1/2.

The power operator requires numeric types and matches the conversion rules for binary arithmetic operators, so in this case it will return either a float or a complex number.

>>> 9 ** (1/2)
3.0
>>> 9 ** .5  # Same thing
3.0
>>> 2 ** .5
1.4142135623730951

(Note: in Python 2, 1/2 is truncated to 0, so you have to force floating point arithmetic with 1.0/2 or similar. See Why does Python give the "wrong" answer for square root?)

This method can be generalized to nth root, though fractions that can't be exactly represented as a float (like 1/3 or any denominator that's not a power of 2) may cause some inaccuracy:

>>> 8 ** (1/3)
2.0
>>> 125 ** (1/3)
4.999999999999999

Edge cases

Negative and complex

Exponentiation works with negative numbers and complex numbers, though the results have some slight inaccuracy:

>>> (-25) ** .5  # Should be 5j
(3.061616997868383e-16+5j)
>>> 8j ** .5  # Should be 2+2j
(2.0000000000000004+2j)

Note the parentheses on -25! Otherwise it's parsed as -(25**.5) because exponentiation is more tightly binding than unary negation.

Meanwhile, math is only built for floats, so for x<0, math.sqrt(x) will raise ValueError: math domain error and for complex x, it'll raise TypeError: can't convert complex to float. Instead, you can use cmath.sqrt(x), which is more more accurate than exponentiation (and will likely be faster too):

>>> import cmath
>>> cmath.sqrt(-25)
5j
>>> cmath.sqrt(8j)
(2+2j)

Precision

Both options involve an implicit conversion to float, so floating point precision is a factor. For example:

>>> n = 10**30
>>> x = n**2
>>> root = x**.5
>>> n == root
False
>>> n - root  # how far off are they?
0.0
>>> int(root) - n  # how far off is the float from the int?
19884624838656

Very large numbers might not even fit in a float and you'll get OverflowError: int too large to convert to float. See Python sqrt limit for very large numbers?

Other types

Let's look at Decimal for example:

Exponentiation fails unless the exponent is also Decimal:

>>> decimal.Decimal('9') ** .5
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
TypeError: unsupported operand type(s) for ** or pow(): 'decimal.Decimal' and 'float'
>>> decimal.Decimal('9') ** decimal.Decimal('.5')
Decimal('3.000000000000000000000000000')

Meanwhile, math and cmath will silently convert their arguments to float and complex respectively, which could mean loss of precision.

decimal also has its own .sqrt(). See also calculating n-th roots using Python 3's decimal module