Question

If a=3+2√2
Find a^3+1/a^3
Next...
a^4+1/a^4

Answer 1

Answer:

a³ + 1/a³ = 198

a⁴ + 1/a⁴ = 1154

Step-by-step explanation:

First, work out

a + 1/a = 3 + 2√2 + 1 / ( 3 + 2√2 )

= 3 + 2√2 + ( 3 - 2√2 ) / ( 9 - 8 )

= 3 + 2√2 + 3 - 2√2

= 6.  (**)

Now square to get

6² = ( a + 1/a )² = a² + 2 a (1/a) + 1/a² = a² + 1/a² + 2,

so a² + 1/a² = 36 - 2 = 34.  (****)

We weren't asked for that one, but that sets the stage.

Cubing (**) give

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

= a³ + 3a + 3/a + 1/a³

= a³ + 1/a³ + 3 ( a + 1/a )

= a³ + 1/a³ + 3 ( 6 ).

Hence

a³ + 1/a³ = 6³ - 18 = 198.

For the last one, use (****) to get

34² = ( a² + 1/a² )² = a⁴ + 2 a² (1/a)² + 1/a⁴

= a⁴ + 2 + 1/a⁴,

so a⁴ + 1/a⁴ = 34² - 2 = 1154.