Question

If x =1-√2, find the value of (x - 1/x)^3​

Answer 1

ANSWER:

Given

• x = 1 - √2

TO FIND:

(x - 1/x)³

By simplifying the question

Using

(a - b)³ = a³ - b³ - 3ab(a - b)

Similarly

(x - 1/x)³ = x³ - (1/x)³ - 3(x)(1/x)[x - 1/x]

(x - 1/x)³ = x³ - 1/x³ - 3(x² - 1/x)

Given x = 1 - √2

Substituting we get

(x - 1/x)³ = (1 - √2)³ - 1/(1 - √2)³ - 3[(1 - √2)² - 1/1 - √2]

Using

(a - b)³ = a³ - b³ - 3ab(a - b)

(a - b)² = a² + b² - 2ab

(x - 1/x)³ = 1³ - (√2)³ - 3(1)(√2)[1 - √2] - 1/1³ - (√2)³ - 3(1)(√2)[1 - √2] - 3[1² + (√2)² - 2(1)(√2) - 1/1 - √2]

(x - 1/x)³ = 1 - 2√2 - 3√2 + 6 - 1/1 - 2√2 - 3√2 + 6 - 3[1 + 2 - 2√2 - 1/1 - √2]

(x - 1/x)³ = 7 - 5√2 - 1/7 - 5√2 - 3[2(1 - √2)/1 - √2]

(x - 1/x)³ = (7 - 5√2)² - 1/7 - 5√2 - 3(2)

Using , (a - b)² = a² + b² - 2ab

(x - 1/x)³ = 7² + (5√2)² - 2(7)(5√2) - 1/7 - 5√2 - 6

(x - 1/x)³ = 48 + 50 - 70√2/7 - 5√2 - 6

(x - 1/x)³ = 98 - 70√2/7 - 5√2 - 6

(x - 1/x)³ = 14(7 - 5√2)/7 - 5√2 - 6

(x - 1/x)³ = 14 - 6

(x - 1/x)³ = 8

Hence,

♦ (x - 1/x)³ = 8