Question

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

Answer 1

Given :-

▪ x = 2 + √3

To Find :-

▪ (x - 1/x)³

Solution :-

Given,

⇒ x = 2 + √3 ...(1)

So, Let's find the value of 1/x,

⇒ 1/x = 1/(2 + √3)

Rationalising the denominator,

⇒ 1/x = { 2 - √3 } / (2 + √3)(2 - √3)

⇒ 1/x = (2 - √3) / (4 - 3)

[∴ (a - b)(a + b) = a² - b² ]

⇒ 1/x = 2 - √3 ...(2)

We have to find the value of (x + 1/x)³, So

Subtracting (2) from (1) to get x - 1/x

⇒ x - 1/x = 2 + √3 - (2 - √3)

⇒ x - 1/x = 2 + √3 - 2 + √3

⇒ x - 1/x = 2√3

Now, Raising to the power of 3,

⇒ (x - 1/x)³ = (2√3)³

⇒ (x - 1/x)³ = 8 × 3 × √3

⇒ (x - 1/x)³ = 24√3

Hence, The value of (x - 1/x)³ is 24√3.

Answer 2

Answer:

given

$x = 2 + \sqrt{3}$

we can Write

$\frac{1}{x} = \frac{1}{2 + \sqrt{3} }$

rationalizing denominator we get

$\frac{1}{x} = \frac{2 - \sqrt{3} }{4 - 3} \\ \\ \frac{1}{x} = 2 - \sqrt{3}$

Now

$x - \frac{1}{x} = 2 + \sqrt{3} - 2 + \sqrt{3} \\ \\ x - \frac{1}{x} = 2\sqrt{3}$

cubing both sides we get

$( {x - \frac{1}{x} })^{3} = ( {2 \sqrt{3} })^{3} \\ \\ ( {x - \frac{1}{x} })^{3 } = 24 \sqrt{3}$