The value of `6+(log)_(3//2)(1/(3sqrt(2))sqrt(sqrt(4-1/(3sqrt(2)))sqrt(4-1/(sqrt(2)))sqrt(4-1/(3sqrt(2)))))` is ...............

Answer

4

Explanation

Plan Use of infinite series
i.e. If `y=sqrt(xsqrt(xsqrt(x ...oo)))rArry=sqrt(xy),`
Given,`6+log_((3)/(2))((1)/(3sqrt(2))sqrt(4-(1)/(3sqrt(2))sqrt(4-(1)/(3sqrt(2))sqrt(4-(1)/(3sqrt(2))...))))`
Let `sqrt(4-(1)/(3sqrt(2))sqrt(4-(1)/(3sqrt(2)))sqrt(...))=y`
`therefore y= sqrt(4-(1)/(3sqrt(2))y)`
`rArr y^(2)+(1)/(3sqrt(2))y-4=0`
`rArr 3sqrt(2)y^(2)+y-12sqrt(2)=0`
`therefore y= (-1pm 17)/(6sqrt(2))`
`or y=(8)/(3sqrt(2))`
Now, `6+log_((3)/(2))((1)/(3sqrt(2))*y)=6+log_((3)/(2))((1)/(3sqrt(2))*(8)/(3sqrt(2)))`
`=6+log_((3)/(2))((4)/(9)) =6+log_((3)/(2))((3)/(2))^(-2)`
`=6-2*log_((3)/(2)) ((3)/(2))=4`