Question

solve questions early dear friendso​

Answer 1

Answer:

$x=\sqrt[3]{2+\sqrt{3} }\\ x^{3}+\frac{1}{x^{3}} = ?$  ...........(1)

We know that

$x=\sqrt[3]{2+\sqrt{3} }$

Transpose ∛ to the other side, it becomes x³.

$x^{3}={2+\sqrt{3} }$............ Substitute this in (1).

The result is:

${2+\sqrt{3} }+\frac{1}{2+\sqrt{3} }$

Rationalise $\frac{1}{2+\sqrt{3} }$.

$2+\sqrt{3} +\frac{1}{2+\sqrt{3}}*\frac{2-\sqrt{3} }{2-\sqrt{3} }$

$2+\sqrt{3} +\frac{2-\sqrt{3} }{(2+\sqrt{3)(2-\sqrt{3})}}$................(2)

We know that (a+b)(a-b)=a²-b²

(2+√3)(2-√3)⇒2²-(√3)³⇒4-3⇒1.

Thus, (2+√3)(2-√3)=1.......Substitute this in (2).

$2+\sqrt{3} +\frac{2-\sqrt{3} }{1}$

$2+\sqrt{3} +{2-\sqrt{3}$......(Rearrange the terms)

$2+2+\sqrt{3}-\sqrt{3}$....................(-√3 and √3 gets cancelled, 2+2=4.)

=4

Thus, (b) is the correct option.

HOPE THIS HELPS :D