If r is positive real number such that `4sqrt(r)-(1)/(4sqrt(r))= 4,` then find the value of ` 6sqrt(r)+(1)/(6sqrt(r)).`

Explanation

We have `4sqrt(r)-(1)/(4sqrt(r))= 4`
On squaring, we get
`sqrt(r)-(1)/(sqrt(r))-2= 16`
or `sqrt(r)-(1)/(sqrt(r))= 18`
Now, let `6sqrt(r)+(1)/(6sqrt(r))= x`
On cubing both sides, we get
`sqrt(r)+(1)/(sqrt(r))+3(6sqrt(r) +(1)/(6sqrt(r))) = x^(3)`
`rArr 18 + 3x = x^(3)`
`rArr x^(3) - 3x - 18 = 0`
`rArr (x - 3) (x^(2) + 3x + 6) = 0`
`rArr x = 3 (as x^(2) + 3x + 6 = 0 has complex roots)`
`therefore 6sqrt(r) +(1)/(6sqrt(r) = 3`