Question

if x = 2 + root 3 then find x^3 + 1/x^3

Answer 1

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if x = 2 + root 3 then find x^3 + 1/x^3

$\huge\tt{\bold{\underline{\underline{Answer᎓}}}}$

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$\bold{ = > \frac{ {x}^{3} + 1} { {x}^{ 3} }}$

$\bold{= > \frac{ {(2 + \sqrt{3} )}^{3} + 1}{ {(2 + \sqrt{3} )}^{3} }}$

Here,this identity is used:

$\bold{\red{ = > {(a + b)}^{3} = {a}^{3} + {b}^{3} + 3ab(a + b)}}$

$\bold{= > \frac{ {(2)}^{3} + {( \sqrt{3}) }^{3} + 3(2)( \sqrt{3} ) (2 + \sqrt{3} ) + 1}{ {(2)}^{3} + {( \sqrt{3}) }^{3} + 3(2)( \sqrt{3} )(2 + \sqrt{3}) }}$

$\bold{= > \frac{4 + 3 + 6 \sqrt{3} (2 + \sqrt{3}) + 1 }{4 + 3 + 6 \sqrt{3} (2 + \sqrt{3} )}}$

$\bold{= > \frac{7 + 6 \sqrt{3} (2 + \sqrt{3}) + 1 }{7 + 6 \sqrt{3} (2 + \sqrt{3}) } = 1}$

$\bold{\pink{ = > 7 + 6 \sqrt{3} (2 + \sqrt{3} )}}$ is being cancelled out as it comes on both numerator as well as denominator.

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