Question

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

Answer 1

Answer:

Proof of $\dfrac{1}{z}}=\dfrac{1}{y}+\dfrac{2}{x}$  is shown below.

Explanation:

The given equation is

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

To prove : $\dfrac{1}{z}=\dfrac{1}{y}+\dfrac{2}{x}$

Proof : Let as assume,

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

$2^x=k\Rightarrow 2=k^{\frac{1}{x}}$    ...(1)

$3^y=k\Rightarrow 3=k^{\frac{1}{y}}$      ...(2)

$12^z=k\Rightarrow 12=k^{\frac{1}{z}}$    ...(3)

We know that

$12=2\times 2\times 3$

$12=(2)^2\times 3$

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

$k^{\frac{1}{z}}=(k^{\frac{1}{x}})^2\times k^{\frac{1}{y}}$

$k^{\frac{1}{z}}=k^{\frac{2}{x}}\times k^{\frac{1}{y}}$

$k^{\frac{1}{z}}=k^{\frac{2}{x}+\frac{1}{y}}$

On comparing both sides, we get

$\dfrac{1}{z}}=\dfrac{2}{x}+\dfrac{1}{y}$

$\dfrac{1}{z}}=\dfrac{1}{y}+\dfrac{2}{x}$

Hence proved.