Question

If x = 1 - root 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 / X = 1 / (1 - √2)

Denominator = 1 - √2

We know that

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

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

On Rationalising the denominator then

=>1/X = 1×(1+√2)/[(1-√2)(1+√2)]

=>1/X = (1+√2)/[1^2-(√2)^2]

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

=>1/X = (1+√2)/(1-2)

=>1/X = (1+√2)/(-1)

=>1/X = -(1+√2)

Now

X -(1/X) = (1-√2)-[-(1+√2)]

=>X -(1/X) = (1-√2)+(1+√2)

=>X - 1/X = 1-√2+1+√2

=>X -1/X = 1+1

=>X - 1/X = 2

Now the value of [X -(1/X)]^3

=>2^3

=>8

Answer:-

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

Used formulae:-

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

• (a+b)(a-b)=a^2-b^2

Used Concept:-

Rationalising factor:-

• The product of two irrational numbers is a rational number then the two numbers are called Rationalising factor of each other.

Example:-The Rationalising factor of

a -√b = a+√b