Assume positive under the square root

[Paul Royik]

Hello.

Is it possible to assume the positiveness of expressions under the square root, so sqrt(x^2)=x or sqrt((x-2)^3)=(x-2)sqrt(x-2)?

Can this be specified fo any expression?

Thank you.

[Pratyksh Gupta]

Hello, 

Yes, you can assume that an expression under a square root is positive using assumptions and simplify(). However, sympy does not automatically assume non-negative values for symbolic expressions unless explicitly specified.

Example 1: Handling sqrt(x^2) = x

You can assume x is positive using the symbols function:

from sympy import symbols, sqrt, simplify 

 x = symbols('x', positive=True) 

expr = sqrt(x**2) 

simplified_expr = simplify(expr) 

print(simplified_expr) # Output: x

If x is not explicitly assumed to be positive, sympy will return sqrt(x^2) = |x| instead.

Example 2: Handling sqrt((x - 2)^3) = (x - 2)*sqrt(x - 2)

You need to ensure that x - 2 is positive:

y = symbols('y', positive=True) # Let y = x - 2 be positive
expr = sqrt(y**3)
simplified_expr = simplify(expr)
print(simplified_expr) # Output: y*sqrt(y)

For an arbitrary expression, you can try using simplify() along with assumptions, but sympy may not always simplify it in the way you expect.

Regards, 
Pratyksh

[emanuel.c...@gmail.com]

It seems not :

>>> from sympy import * >>> x=symbols("x") >>> with assuming(Q.positive(x)): sqrt(x**2).simplify() ... sqrt(x**2) >>> with assuming(Q.positive(x)): sqrt(x**2).factor() ... sqrt(x**2) >>> with assuming(Q.positive(x)): sqrt(x**2).expand() ... sqrt(x**2)

But :

>>> y=symbols("y", positive=True) >>> sqrt(y**2) y

This is a bit inconsistent…

​

[Krishnav Bajoria]

Hello Paul,

Expanding on the above answer ,there are a few more different ways to assume the positiveness of expressions under the square root in SymPy (using the new assumptions) . However, it seems that we still cannot directly simplify terms of the form x**(3/2) to x * sqrt(x), if that is the intended goal. That said, expressions where the exponent in the numerator is even can indeed be reduced, as demonstrated in the following examples:

from sympy import *

x = symbols('x')

with assuming(Q.positive(x)): # all assumptions can be put within assuming () 
    print(refine(sqrt(x**2)))  # Output: x

# Using refine syntax of type refine(expr, assumptions)
print(refine(sqrt(x**2), Q.positive(x)))  # Output: x  

# A few more examples:
print(refine(sqrt(x**3), Q.positive(x)))  # Output: x**(3/2)
print(refine(sqrt(x**4), Q.positive(x)))  # Output: x**2
print(refine(sqrt(x**5), Q.positive(x)))  # Output: x**(5/2)

This approach effectively simplifies square roots for even exponents, while odd exponents retain their fractional form.

That said, if anyone in the community knows a way to express odd exponents (e.g., y**(3/2)) in the form of y * sqrt(y), it would be nice to know that!

Best regards,

Krishnav Bajoria.

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