Question

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

Answer 1

Answer:

Friend please see in the question in first term x=1-√2 you have put the min (-) sigh there will be plus(+) sigh because answer is not coming correct.

Step-by-step explanation:

Given:-

x = 1 + √2

To find out:-

Value of (x - 1/x)^3

Solution:-

We have,

x = 1 + √2

∴ 1/x = 1/1+√2

The denomination = 1 + √2

We know that

Rationalising factor of a+√b = a-√b

So, the rationalising factor of 1+√2 = 1-√2

On rationalising the denominator them

1/x = [1/(1+√2)]×[(1-√2)/1-√2)]

1/x = [1(1-√2)]/[(1+√2)(1-√2)]

1/x = (1-√2)/[(1+√2)(1-√2)]

Now, we will apply algebraic Identity in denominator because the denominator is in the form of

(a+b)(a-b) = a^2 - b^2

Where we have to put a = 1 and b = √2

1/x = (1-√2)/[(1)^2 - (√2)^2]

1/x = (1-√2)/(1 - 2)

1/x = (1-√2)/-1

1/x = √2+1

Now, we have to Sub.. both values x and 1/x, we get

∴ x - 1/x = 1+√2-√2+1

√2 will be cancel out

x - 1/x = 1 + 1

x - 1/x = 2

Now, cubing on both sides, we get

→ (x - 1/x)^3 = (2)^3

Now, applying algebraic Identity because, our expression in the form of; (a+b)^3 = a^3-b^3-3ab(a-b)

Where, we have to put a = x and b = 1/x. we get

→ x^3 - 1/x^3 - 3(x)(1/x) - (x - 1/x) = 8

→ x^3 - 1/x^3 - 3(x - 1/x) = 8

→ x^3 - 1/x^3 - 3(2) = 8

→ x^3 - 1/x^3 - 6 = 8

→ x^3 - 1/x^3 = 8+6

→ x^3 - 1/x^3 = 14.