Question

If a+1/a=3
What is the answer? a⁵+1/a⁵
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Answer 1

123

Explanation:

We can square and cube the equation, then multiply the results together.

(a +
`(1)/(a)`) = 3 ← square both sides

a² + 2 +
`(1)/(a^2)` = 9 ( subtract 2 from both sides )

a² +
`(1)/(a^2)` = 7 → (1)

-----------------------

Now cube both sides

a³ +
`(1)/(a^3)` + 3(a +
`(1)/(a)`) = 27

a³ +
`(1)/(a^3)` + 3(3) = 27

a³ +
`(1)/(a^3)` + 9 = 27 ( subtract 9 from both sides )

a³ +
`(1)/(a^3)` = 18 → (2)

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Multiply (1) and (2)

(a² +
`(1)/(a^2)` )(a³ +
`(1)/(a^3)` ) = 7(18)

`a^(5)` + (
`(1)/(a)` + a) +
`(1)/(a^5)` = 126

`a^(5)` + 3 +
`(1)/(a^5)` = 126 ( subtract 3 from both sides )

`a^(5)` +
`(1)/(a^5)` = 123