Question

If a^x = b^y = c^z and abc = 1, show that

`(1)/x + (1)/y + (1)/z` = 0.

Answer 1

a^x = b^y = c^z 

So, a^x = b^y ⇒ a =`"b"^(y/x)      .....("Using" "a"^(1/"n") = root("n")("a"))`

b^y = c^z ⇒ c = `"b"^(y/z)      .....("Using" "a"^(1/"n") = root("n")("a"))`

and abc = 1

⇒ `"b"^(y/x) · "b"·"b"^(y/z)` = 1

⇒ `"b"^(y/x) · "b"·"b"^(y/z)` = 1

⇒ `"b"^(y/x + 1 + y/z)` = b°   ......(Using a° = 1)

⇒ `y/x + 1 + y/z` = 0
Divide throughout by y.
⇒ `(1)/x + (1)/y + (1)/z` = 0
Hence proved.