Question

If 2^x=3^y=12^z, show that 1÷z=1÷y+2÷x.....​

Answer 1

Proof of
`(1)/(z)}=(1)/(y)+(2)/(x)` is shown below.

Step-by-step explanation:

The given equation is

`2^x=3^y=12^z`

To prove :
`(1)/(z)=(1)/(y)+(2)/(x)`

Proof : Let as assume,

`2^x=3^y=12^z=k`

`2^x=k\Rightarrow 2=k^{(1)/(x)}` ...(1)

`3^y=k\Rightarrow 3=k^{(1)/(y)}` ...(2)

`12^z=k\Rightarrow 12=k^{(1)/(z)}` ...(3)

We know that

`12=2* 2* 3`

`12=(2)^2* 3`

Using (1), (2) and (3), we get

`k^{(1)/(z)}=(k^{(1)/(x)})^2* k^{(1)/(y)}`

`k^{(1)/(z)}=k^{(2)/(x)}* k^{(1)/(y)}`

`k^{(1)/(z)}=k^{(2)/(x)+(1)/(y)}`

On comparing both sides, we get

`(1)/(z)}=(2)/(x)+(1)/(y)`

`(1)/(z)}=(1)/(y)+(2)/(x)`

Hence proved.