# Fungrim entry: 73b76c

$\sqrt{a b} = \sqrt{a} \sqrt{b}$
Assumptions:$\left(a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \left[0, \infty\right)\right) \;\mathbin{\operatorname{or}}\; \left(b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \in \left[0, \infty\right)\right)$
Alternative assumptions:$a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \arg(a) + \arg(b) \in \left(-\pi, \pi\right]$
TeX:
\sqrt{a b} = \sqrt{a} \sqrt{b}

\left(a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \left[0, \infty\right)\right) \;\mathbin{\operatorname{or}}\; \left(b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \in \left[0, \infty\right)\right)

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \arg(a) + \arg(b) \in \left(-\pi, \pi\right]
Definitions:
Fungrim symbol Notation Short description
Sqrt$\sqrt{z}$ Principal square root
CC$\mathbb{C}$ Complex numbers
ClosedOpenInterval$\left[a, b\right)$ Closed-open interval
Infinity$\infty$ Positive infinity
Arg$\arg(z)$ Complex argument
OpenClosedInterval$\left(a, b\right]$ Open-closed interval
Pi$\pi$ The constant pi (3.14...)
Source code for this entry:
Entry(ID("73b76c"),
Formula(Equal(Sqrt(Mul(a, b)), Mul(Sqrt(a), Sqrt(b)))),
Variables(a, b),
Assumptions(Or(And(Element(a, CC), Element(b, ClosedOpenInterval(0, Infinity))), And(Element(b, CC), Element(a, ClosedOpenInterval(0, Infinity)))), And(Element(a, CC), Element(b, CC), Element(Add(Arg(a), Arg(b)), OpenClosedInterval(Neg(Pi), Pi)))))

## Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC