Question

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

Answer 1

Step-by-step explanation:

Given :-

x = 1-√2

To find :-

Find the value of [x-(1/x)]³ ?

Solution :-

Given that

x = 1-√2 ------------(1)

=> 1/x = 1/(1-√2)

The denominator = 1-√2

The Rationalising factor of 1-√2 is 1+√2

On Rationalising the denominator then

=> 1/x = [1/(1-√2)]×[(1+√2)/(1+√2)]

=> 1/x = (1+√2)/[(1-√2)(1+√2)]

=> 1/x = (1+√2)/[1²-(√2)²]

Since , (a+b)(a-b) = a²-b²

=> 1/x = (1+√2)/(1-2)

=> 1/x =(1+√2)/(-1)

=> 1/x = -(1+√2) ------------(2)

On Subtracting (2) from (1) then

=> x-(1/x)

=> (1-√2)-(-(1+√2))

=> (1-√2)+(1+√2)

=> 1-√2+1+√2

=> (1+1)+(√2-√2)

=> 2+0

=> 2

The value of x-(1/x) = 2

Now,

The value of of [x-(1/x)]³

=> 2³

=>2×2×2

=> 8

Answer:-

The value of of [x-(1/x)]³ for the given problem is 8

Used formulae:-

→ (a+b)(a-b) = a²-b²

→ The Rationalising factor of a-√b = a+√b