Combine sqrt?

[Paul Royik]

What is the best way to convert sqrt(x)*sqrt(x-4) to sqrt(x^2-4x) or (x^2+5x+4)/sqrt(x)/sqrt(x-4) to (x^2+5x+4)/sqrt(x^2-4x)

I tried replace, but it doesn't work in second case.

[John Peterson]

On Thursday, February 26, 2015 at 8:32:11 AM UTC-7, Paul Royik wrote:

What is the best way to convert sqrt(x)*sqrt(x-4) to sqrt(x^2-4x) or (x^2+5x+4)/sqrt(x)/sqrt(x-4) to (x^2+5x+4)/sqrt(x^2-4x)

I tried replace, but it doesn't work in second case.

Be careful, they aren't equal for all values of x, but subs() will do this if you are willing to be explicit...

#!/usr/bin/env python
from sympy import *
print sympify('(x**2 + 5*x + 4) / sqrt(x) / sqrt(x-4)').subs(sympify('sqrt(x)*sqrt(x-4)'), sympify('sqrt(x**2 - 4*x)'))

Output:

(x**2 + 5*x + 4)/sqrt(x**2 - 4*x) 

 

[Ondřej Čertík]

Hi Paul,

How did you use replace?

Note that this conversion is only valid for some "x", not all complex
"x", for example if you use imaginary unit for "x":

In [14]: x = I

In [15]: sqrt(x)*sqrt(x - 4)
Out[15]: (-1)**(1/4)*sqrt(-4 + I)

In [16]: (sqrt(x)*sqrt(x - 4)).n()
Out[16]: -1.24962106768765 + 1.60048518044024*I

In [17]: sqrt(x**2 - 4*x)
Out[17]: sqrt(-1 - 4*I)

In [18]: sqrt(x**2 - 4*x).n()
Out[18]: 1.24962106768765 - 1.60048518044024*I

Ondrej

[Paul Royik]

I have assumption that x is positive.

a, b = Wild...

I use like f.replace(sqrt(a)*sqrt(b), sqrt((a*b).expand()))

[Paul Royik]

I need something more general

[Aaron Meurer]

See http://docs.sympy.org/latest/tutorial/simplification.html#powers.
powsimp(expr, force=True) will do what you want.

Aaron Meurer

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[Paul Royik]

It doesn't work if x is positive.

In fact, when x is positive, 1/sqrt(x*(x-4)) automatically converted to 1/sqrt(x)/sqrt(x-4)

[Ondřej Čertík]

On Thu, Feb 26, 2015 at 4:05 PM, Paul Royik <distan...@gmail.com> wrote:
> It doesn't work if x is positive.

Indeed, looks like a bug. As a workaround, you can always substitute
general "x" for the positive "x", then it will work:

In [4]: x = Symbol("x", positive=True)

In [5]: powsimp(sqrt(x)*sqrt(x-4), force=True)
Out[5]:
___ _______
╲╱ x ⋅╲╱ x - 4

In [6]: powsimp((sqrt(x)*sqrt(x-4)).subs(x, Symbol("x"), force=True)
...:
KeyboardInterrupt

In [6]: powsimp((sqrt(x)*sqrt(x-4)).subs(x, Symbol("x")), force=True)
Out[6]:
___________
╲╱ x⋅(x - 4)

> In fact, when x is positive, 1/sqrt(x*(x-4)) automatically converted to
> 1/sqrt(x)/sqrt(x-4)

The same trick:

In [7]: powsimp(1/(sqrt(x)*sqrt(x-4)).subs(x, Symbol("x")), force=True)
Out[7]:
1
─────────────
___________
╲╱ x⋅(x - 4)


Ondrej

> https://groups.google.com/d/msgid/sympy/dc099057-8529-4f12-ae42-ed7f9e89b98e%40googlegroups.com.

[Paul Royik]

Thank you.

Got it!

[Aaron Meurer]

Ah, apparently the core is looking at assumptions and automatically
simplifies based on them. I don't think that any automatic
simplification should happen based on assumptions.

Aaron Meurer

> https://groups.google.com/d/msgid/sympy/b9754512-debb-4099-bfab-3724b3be1568%40googlegroups.com.

[Paul Royik]

I agree.