Verified answer

Solution :

1) Given, x = 1 - √2

To find : (x + 1/x)²

Now, 1/x = 1/(1 - √2)

Rationalizing the denominator,

→ 1/x = (1 + √2)/[(1 + √2)(1 - √2)]

→ 1/x = (1 + √2)/[(1)² - (√2)²]

→ 1/x = (1 + √2)/(1 - 2)

→ 1/x = (1 + √2)/(-1)

→ 1/x = -(1 + √2)

→ 1/x = - 1 - √2

Now,

→ (x + 1/x)² = (1 - √2 - 1 - √2)²

→ (x + 1/x)² = (-2√2)²

→ (x + 1/x)² = 4 × 2

→ (x + 1/x)² = 8 (Ans.)

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2) Given, x⁴ + 1/x⁴ = 119

To find : x - 1/x & x³ - 1/x³

Now,

→ x⁴ + 1/x⁴ = 119

→ (x²)² + (1/x²)² = 119

→ (x² + 1/x²)² - 2 = 119

→ (x² + 1/x²)² = 121

→ x² + 1/x² = √121

→ x² + 1/x² = 11

→ (x - 1/x)² + 2 = 11

→ (x - 1/x)² = 9

→ x - 1/x = ±√9

→ x - 1/x = ±3 (Ans.)

Again,

→ x³ - 1/x³ = (x - 1/x)³ + 3 × x × 1/x (x - 1/x)

→ x³ - 1/x³ = (±3)³ + 3(±3)

→ x³ - 1/x³ = ±27 ±9

→ x³ - 1/x³ = ±36 (Ans.)

Answer:

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