Question

If `9"a"^2 + (1)/(9"a"^2) = 23`; find the value of `27"a"^3 + (1)/(27"a"^3)`

Answer 1

`9"a"^2 + (1)/(9"a"^2) = 23`

Using `(3"a" + 1/(3"a"))^2`

= `(3"a")^2 + (1/(3"a"))^2 + 2(3"a") (1/(3"a"))`

⇒ `(3"a" + 1/(3"a"))^2`

= `9"a"^2 + 1/(9"a"^2) + 2`
= 23 + 2
= 25

⇒ `3"a" + 1/(3"a")` = 5
Cubing both sides, we get :

`(3"a")^3 + (1/(3"a"))^3 + 3(3"a") (1/(3"a")) (3"a" + 1/(3"a"))` = (5)³

⇒ `27"a"^3 + 1/(27"a"^3) + 3(5)` = 125

⇒ `27"a"^3 + 1/(27"a"^3)`
= 125 - 15
= 110.