Question

if x=2+√3, then find the value of x^3+1/x^3​

Answer 1

$\color {red}\huge\bold\star\underline\mathcal{Question:-}$ we have to Find x³ + 1/x³ ?

$\huge\boxed{\fcolorbox{cyan}{grey}{Solution:--}}$

x = 2+√3 (Given)

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

so,

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

$x \times \frac{1}{x} = 1$

Now we know that ,

$\large\red{\boxed{\sf </strong><strong>{a}^{3}</strong><strong>\</strong><strong>:</strong><strong>+</strong><strong>\</strong><strong>:</strong><strong>{b}^{3}</strong><strong>\</strong><strong>:</strong><strong>=</strong><strong>\</strong><strong>:</strong><strong>(a + b)^{3} - </strong><strong>\</strong><strong>:</strong><strong>3ab(a + b)</strong><strong>}}$

Putting all values we get,

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

$\bold{\boxed{\boxed{ADDITIONAL\:BRAINLY\:INFORMATION\:-:}}}$

[1] ( a + b )² = a² + 2ab + b²

[2] ( a – b )² = a² – 2ab + b²

[3] ( a + b )³ = a³ + 3a² b + 3ab² + b³

[4] ( a – b )³ = a ³ – 3a² b + 3ab² – b³

[5] ( a + b )( a ² - ab + b² ) = a³ + b³

[6] ( a – b )( a ² + ab + b² ) = a³ – b³