Question

x=2+√3, find x^4+1/x^4​

Answer 1

EXPLANATION.

⇒ x = 2 + √3.

As we know that,

We can write equation as,

⇒ 1/x = 1/(2 + √3).

Rationalizes the denominator in the equation, we get.

⇒ 1/x = 1/(2 + √3) x (2 - √3)/(2 - √3).

⇒ 1/x = (2 - √3)/[(2 + √3)(2 - √3)].

⇒ 1/x = (2 - √3)/[(2)² - (√3)²].

⇒ 1/x = (2 - √3)/[4 - 3].

⇒ 1/x = (2 - √3).

We can write equation as,

⇒ x + 1/x = [(2 + √3) + (2 - √3)].

⇒ x + 1/x = 2 + √3 + 2 - √3.

⇒ x + 1/x = 4.

Squaring on both sides of the equation, we get.

⇒ (x + 1/x)² = (4)².

As we know that,

Formula of :

⇒ (a + b)² = a² + b² + 2ab.

Using this formula in the equation, we get.

⇒ (x)² + (1/x)² + 2(x)(1/x) = (4)².

⇒ x² + 1/x² + 2 = 16.

⇒ x² + 1/x² = 16 - 2.

⇒ x² + 1/x² = 14.

Again Squaring on both sides of the equation, we get.

⇒ [(x² + 1/x²)]² = (14)².

⇒ (x²)² + (1/x²)² + 2(x²)(1/x²) = (14)².

⇒ x⁴ + 1/x⁴ + 2 = 196.

⇒ x⁴ + 1/x⁴ = 196 - 2.

⇒ x⁴ + 1/x⁴ = 194.