Question

If a+ 1/a = 5 then find the value of; i) a² + 1/a² ii) a³ + 1/a³​

Answer 1

23 and 110

Explanation:

using the expansion

(a + b)² = a² + 2ab + b² , then

a +
`(1)/(a)` = 5 ( square both sides )

(a +
`(1)/(a)` )² = 5²

a² + 2(a ×
`(1)/(a)` ) +
`(1)/(a^2)` = 25

a² + 2(1) +
`(1)/(a^2)` = 25

a² + 2 +
`(1)/(a^2)` = 25 ( subtract 2 from both sides )

a² +
`(1)/(a^2)` = 23

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using the expansion

(a + b)³ = a³ + b³ + 3ab(a + b) , then

a +
`(1)/(a)` = 5 ( cube both sides )

(a +
`(1)/(a)` )³ = 5³

a³ +
`(1)/(a^3)` + 3(a ×
`(1)/(a)` )(a +
`(1)/(a)` ) = 125

a³ +
`(1)/(a^3)` + 3(1)(5) = 125

a³ +
`(1)/(a^3)` + (3 × 5) = 125

a³ +
`(1)/(a^3)` + 15 = 125 ( subtract 15 from both sides )

a³ +
`(1)/(a^3)` = 110

Answer 2

23

110

Explanation:

We have a + 1/a = 5

(a + 1/a)² = a² + (1/a²) + 2.a.1/a = a² + (1/a²) + 2

But (a + 1/a)² = 5² = 25

So a² + (1/a²) = 25 - 2 = 23

(a + 1/a)³ = a³ + 3a²(1/a) + 3a(1/a)² + (1/a)2

= a³ + 1/a³ + 3a + 3/a

= a³ + 1/a³ + 3(a + 1/a)

= a³ + 1/a³ + 3(5)

= a³ + 1/a³ + 15

But (a + 1/a)³ = 5³ = 125

So

a³ + 1/a³ + 15 = 125

a³ + 1/a³ = 110