Question

Ifx + 1/x= 6, find the value of x3 -1/x3​

Answer 1

$\sf given \: x + \frac{1}{x} = 6$

cubing on both sides

$\sf \: {(x + \frac{1}{x} )}^{3} = {6}^{3}$

$\sf \: { {x}^{3} + \frac{1}{{x}^{3}}} + 3(x + \frac{1}{x}) = 216$

$\sf \: { {x}^{3} + \frac{1}{{x}^{3}}} + 3(6) = 216$

$\sf \: { {x}^{3} + \frac{1}{{x}^{3}}} + 18 = 216$

$\sf \: { {x}^{3} + \frac{1}{{x}^{3}}} = 216 - 18$

$\fbox{\sf \: { {x}^{3} + \frac{1}{{x}^{3}}} = 198}$

$\huge\blue{\ddot\smile}$

Answer 2

EXPLANATION.

⇒ (x + 1/x) = 6.

As we know that.

Squaring on both sides of the equation, we get.

⇒ (x + 1/x)² = (6)².

⇒ x² + 1/x² + 2(x)(1/x) = 36.

⇒ x² + 1/x² + 2 = 36.

⇒ x² + 1/x² = 36 - 2.

⇒ x² + 1/x² = 34.

As we know that,

Formula of :

⇒ (x - 1/x)² = x² + 1/x² - 2(x)(1/x).

⇒ (x - 1/x)² = x² + 1/x² - 2.

Put the value of x² + 1/x² = 34 in the equation, we get.

⇒ (x - 1/x)² = 34 - 2.

⇒ (x - 1/x)² = 32.

⇒ (x - 1/x) = √32.

⇒ (x - 1/x) = 4√2.

As we know that,

⇒ (x - 1/x) = 6.

Cubing on both sides of the equation, we get.

⇒ (x - 1/x)³ = (6)³.

⇒ x³ - 3(x)²(1/x) + 3(x)(1/x)² - 1/x³ = 216.

⇒ x³ - 1/x³ - 3x + 3/x = 216.

⇒ x³ - 1/x³ - 3(x - 1/x) = 216.

Put the value of x - 1/x = 4√2 in the equation, we get.

⇒ x³ - 1/x³ - 3(4√2) = 216.

⇒ x³ - 1/x³ - 12√2 = 216.

⇒ x³ - 1/x³ = 216 + 12√2.