Question

Answer this with clear steps.

Answer 1

a = 2 + √3

⇒ $\frac{1}{a} = \frac{1}{2 + \sqrt{3}}$

Multiplying and Dividing $\frac{1}{(2 + \sqrt{3})}$ with 2 - √3

$\frac{1}{a} = \frac{(2 - \sqrt{3})} {(2 + \sqrt{3}) ( 2 - \sqrt{3})}$

As (P + Q) × (P - Q) = P² - Q²

⇒ (2 + √3) × (2 - √3) = 2² - (√3)² = 4 - 3 = 1

⇒ 1/a = 2 - √3

⇒ a + 1/a = (2 + √3) + (2 - √3) = 4

As (P + Q)³ = P³ + Q³ + 3PQ(P + Q)

⇒ P³ + Q³ = (P + Q)³ - 3PQ(P + Q)

Substituting P = a and Q = 1/a

⇒ a³ + 1/a³ = (a + 1/a)³ - 3 × a × 1/a × (a + 1/a)

⇒ a³ + 1/a³ = 4³ - 3(4) = 64 - 12 = 52

As (P + Q)² = P² + Q² + 2PQ

Substituting P = a and Q = 1/a

⇒ (a + 1/a)² = a² + 1/a² + 2 × a × 1/a

⇒ a² + 1/a² = (a + 1/a)² - 2

⇒ a² + 1/a² = 4² - 2 = 16 - 2 = 14