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

Question

asked Apr 24, 2021 85.3k views

1 Answer

Alturkovic · Answer 1 · 2021-04-29T10:02:29+0000

Answer:

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.