Sqrt[(a + z)/(b - z)] == Sqrt[(a + z)/(a + b)] Sqrt[(a + b)/(b - z)] /; Element[a + b, Reals] && a + b != 0

Cell[BoxData[RowBox[List[RowBox[List[SqrtBox[FractionBox[RowBox[List["a", "+", "z"]], RowBox[List["b", "-", "z"]]]], "\[Equal]", RowBox[List[SqrtBox[FractionBox[RowBox[List["a", "+", "z"]], RowBox[List["a", "+", "b"]]]], SqrtBox[FractionBox[RowBox[List["a", "+", "b"]], RowBox[List["b", "-", "z"]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["a", "+", "b"]], "\[Element]", "Reals"]], "\[And]", RowBox[List[RowBox[List["a", "+", "b"]], "\[NotEqual]", "0"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msqrt> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mi> z </mi> </mrow> <mrow> <mi> b </mi> <mo> - </mo> <mi> z </mi> </mrow> </mfrac> </msqrt> <mo> &#10869; </mo> <mrow> <msqrt> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mi> z </mi> </mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mrow> <mi> b </mi> <mo> - </mo> <mi> z </mi> </mrow> </mfrac> </msqrt> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mo> &#8800; </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> a </ci> <ci> z </ci> </apply> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> a </ci> <ci> z </ci> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> <reals /> </apply> <apply> <neq /> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SqrtBox[FractionBox[RowBox[List["a_", "+", "z_"]], RowBox[List["b_", "-", "z_"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SqrtBox[FractionBox[RowBox[List["a", "+", "z"]], RowBox[List["a", "+", "b"]]]], " ", SqrtBox[FractionBox[RowBox[List["a", "+", "b"]], RowBox[List["b", "-", "z"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["a", "+", "b"]], "\[Element]", "Reals"]], "&&", RowBox[List[RowBox[List["a", "+", "b"]], "\[NotEqual]", "0"]]]]]]]]]]

2001-10-29