Question

x= 2+✓3 find (x^3+1/x^3)​

Answer 1

Answer:

$x = 2 + \sqrt{3} \\ \frac{1}{x} = \frac{1}{2 + \sqrt{3} } \times \frac{2 - \sqrt{3} }{2 - \sqrt{3} } = \frac{2 - \sqrt{3} }{4 - 3} \\ \: \: \: \: \: = \frac{2 - \sqrt{3} }{1} = 2 - \sqrt{3} \\ x + \frac{1}{x} = 2 + \sqrt{3} + 2 - \sqrt{3} = 4 \\( x + \frac{1}{x} ) {}^{3} = 4 {}^{3} \\ x {}^{3} +( \frac{1}{x} ) {}^{3} + 3 \times x \times \frac{1}{x} (x + \frac{1}{x} ) = 64 \\ x {}^{3} + \frac{1}{x {}^{3} } = 64 - 12 = 52$

Answer 2

Answer:

x³ + 1/x³ = 52

Note:

★ (a + b)² = a² + 2ab + b²

★ (a - b)² = a² - 2ab + b²

★ a² - b² = (a + b)(a - b)

★ (a + b)³ = a³ + b³ + 3ab(a + b)

★ (a - b)³ = a³ - b³ - 3ab(a - b)

★ a³ + b³ = (a + b)(a² - ab + b²)

Solution:

Given : x = 2 + √3

To find : x³ + 1/x³

We have ;

x = 2 + √3

Thus,

1/x = 1/(2 + √3)

Now,

Rationalising the denominator of the term in RHS , we get ;

=> 1/x = (2 - √3)/(2 + √3)(2 - √3)

=> 1/x = (2 - √3)/[ 2² - (√3)² ]

=> 1/x = (2 - √3)/(4 - 3)

=> 1/x = (2 - √3)/1

=> 1/x = 2 - √3

Now,

=> x + 1/x = 2 + √3 + 2 - √3

=> x + 1/x = 4

Cubing both the sides , we have ;

=> (x + 1/x)³ = 4³

=> x³ + 1/x³ + 3•x•(1/x)•(x + 1/x) = 64

=> x³ + 1/x³ + 3(x + 1/x) = 64

=> x³ + 1/x³ + 3•4 = 64 { ° . ° x + 1/x = 4 }

=> x³ + 1/x³ + 12 = 64

=> x³ + 1/x³ = 64 - 12

=> x³ + 1/x³ = 52

Hence,

The required answer is 52 .