Question

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

Answer 1

Given:-

• x = 2 - √3

To Find:-

• The value of (x - 1/x)³

Solution:-

As we have the value of x = 2 - √3, we can find the value of 1/x too.

Hence,

• 1/x = 1/(2 - √3)

By rationalizing the denominator,

1/x = (2 - √3)/(2 - √3)(2 + √3)

We know,

• (a - b)(a + b) = a² - b²

Hence,

= 1/x = (2 + √3)/[(2)² - (√3)²]

⇒ 1/x = (2 + √3)/(4 - 3)

⇒ 1/x = (2 + √3)/1

⇒ 1/x = 2 + √3

• ∴ We got the value of 1/x as 2 + √3

Now,

We need to find:-

(x - 1/x)³

Putting the values,

[(2 - √3) - (2 + √3)]³

= [2 - √3 - 2 - √3]³

= [-2√3]³

= -2√3 × -2√3 × -2√3

= -24√3

• Therefore the value of x - 1/x is -24√3.

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How to solve?

↦ Firstly we had the value of x = 2 - √3 from where we got 1/x as 1/(2 - √3). By rationalising the denominator we got the value of 1/x as 2 + √3. Then putting the respective values in (x - 1/x)³, we got the required answer.

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