If a = 3 + 2√2, then find the value of (a⁶ – a⁴ – a² + 1)/a³.

Question

Question

Answer 1

Given:

a = 3 + 2√2

Concept Used:

a² – b² = (a – b)(a + b)

a³ + b³ = (a + b)³ – 3ab(a + b)

Calculation:

a = 3 + 2√2

1/a = 1/(3 + 2√2)

⇒ 1/a = (3 – 2√2)/{(3 + 2√2) × (3 – 2√2)}

⇒ 1/a = (3 – 2√2)/{3² – (2√2)²}

⇒ 1/a = (3 – 2√2)/(9 – 8)

⇒ 1/a = (3 – 2√2)

Now,

a + 1/a = 3 + 2√2 + 3 – 2√2

⇒ a + 1/a = 6

(a⁶ – a⁴ – a² + 1)/a³

⇒ a³ – a – 1/a + 1/a³

⇒ (a³ + 1/a³) – (a + 1/a)

⇒ {(a + 1/a)³ – 3(a + 1/a)} – (a + 1/a)

⇒ (6³ – 3 × 6) – 6

⇒ 216 – 18 – 6

⇒ 192

∴ The required value of (a⁶ – a⁴ – a² + 1)/a³ is 192