Answer:
Proof of [tex]\dfrac{1}{z}}=\dfrac{1}{y}+\dfrac{2}{x}[/tex] is shown below.
Explanation:
The given equation is
[tex]2^x=3^y=12^z[/tex]
To prove : [tex]\dfrac{1}{z}=\dfrac{1}{y}+\dfrac{2}{x}[/tex]
Proof : Let as assume,
[tex]2^x=3^y=12^z=k[/tex]
[tex]2^x=k\Rightarrow 2=k^{\frac{1}{x}}[/tex] ...(1)
[tex]3^y=k\Rightarrow 3=k^{\frac{1}{y}}[/tex] ...(2)
[tex]12^z=k\Rightarrow 12=k^{\frac{1}{z}}[/tex] ...(3)
We know that
[tex]12=2\times 2\times 3[/tex]
[tex]12=(2)^2\times 3[/tex]
Using (1), (2) and (3), we get
[tex]k^{\frac{1}{z}}=(k^{\frac{1}{x}})^2\times k^{\frac{1}{y}}[/tex]
[tex]k^{\frac{1}{z}}=k^{\frac{2}{x}}\times k^{\frac{1}{y}}[/tex]
[tex]k^{\frac{1}{z}}=k^{\frac{2}{x}+\frac{1}{y}}[/tex]
On comparing both sides, we get
[tex]\dfrac{1}{z}}=\dfrac{2}{x}+\dfrac{1}{y}[/tex]
[tex]\dfrac{1}{z}}=\dfrac{1}{y}+\dfrac{2}{x}[/tex]
Hence proved.
