Question

Prove the following:

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

Answer 1

L.H.S.

= `root("ab")(x^"a"/x^"b")·root("bc")(x^"b"/x^"c")·root("ca")(x^"c"/x^"a")` 

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

= `x^(1/"b")/(x1/"a")·x^(1/"c")/(x1/"b")·x^(1/"a")/(x1/"c")`  .....(Using (a^m)ⁿ = a^mn)

= `x^(1/"b"-1/"a")·x^(1/"c"-1/"b")·x^(1/"a"-1/"c")`   ....(Using a^m ÷ aⁿ = a^m-n)

= `x^(("a"-"b")/"ab").x^(("b"-"c")/"bc").x^(("c"-"a")/"ac")` 

= `x^(("a-b")/("ab")+("b-c")/("bc")+("c-a")/("ac")`  ....(Using a^m x aⁿ = a^m+n)

= `x^(("ac"-"bc"+"ab"-"ac"+"bc"-"ab")/"abc"`

= `x^(0/"abc")`
= x⁰ 
= 1         ......(Using a⁰ = 1)
=R.H.S.
Hence proved.