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