Question

Prove that:

`((x^a)/x^b)^(1/(ab))((x^b)/x^c)^(1/(bc))(x^c/x^a)^(1/(ca))=1`

Answer 1

`(("x"^"a")/"x"^"b")^(1/("a""b"))(("x"^"b")/"x"^"c")^(1/("b""c"))("x"^"c"/"x"^"a")^(1/("c""a"))=1`

`"L"."H"."S".=``(("x"^"a")/"x"^"b")^(1/("a""b"))(("x"^"b")/"x"^"c")^(1/("b""c"))("x"^"c"/"x"^"a")^(1/("c""a"))`

`("x"^("a"-"b"))^(1/("a""b"))("x"^("b"-"c"))^(1/("b""c"))("x"^("c"-"a"))^(1/("c""a"))`

`="x"^(("a"-"b")/("a""b"))"x"^(("b"-"c")/("b""c"))"x"^(("c"-"a")/("c""a"))`    .....`{("x"^"a")^"b"="x"^("a""b")}`

`="x"^(("a"-"b")/("a""b")+("b"-"c")/("b""c")+("c"-"a")/("c""a")`

`="x"^(("a""c"-"b""c"+"a""b"-"a""c"+"b""c"-"a""b")/("a""b"""c")`

c(a−b) + a(b−c) + b(c−a) = 0.

`0/(abc) = 0`

`="x"^0=1="R"."H"."S".  ` .....`(∵"x"^0=1)`