Question

`a^x = b^(1/2), b^y = c^(1/3), c^z = a^(1/2)`. What is the value of `xyz`?

Answer 1

`1/12`

Explanation:

Applying `log` to the equations

`xloga=1/2logb`
`ylogb=1/3logc`
`zlogc=1/2loga`

then

`xyz=1/12( logb logc log a)/(loga logb logc)=1/12`

Answer 2

`a^x=b^(1/2)`

`=>a^(xy)=(b^y)^(1/2)=(c^(1/3))^(1/2)=c^(1/6)`

`=>(a^(xy))^z=(c^z)^(1/6)=(a^(1/2))^(1/6)=a^(1/12)`

`=>a^(xyz)=a^(1/12)`

`=>xyz=1/12`