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\!\left(a\right) + \arg\!\left(b\right) \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\!\left(a\right) + \arg\!\left(b\right) \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`	$\arg\!\left(z\right)$	Complex argument
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`ConstPi`	$\pi$	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(ConstPi), ConstPi)))))

Topics using this entry

Square roots