Question

if x= 2+√3 find the value of xpower 3 +1/xpower 3

Answer 1

Given that

$x = 2 + \sqrt{3} \\ \\ \frac{1}{x} = \frac{1}{2 + \sqrt{3} } = \frac{2 - \sqrt{3} }{ {2}^{2} - {(( \sqrt{3)} }^{2}) } \\ \\ = \frac{2 - \sqrt{3} }{4 - 3} = 2 - \sqrt{3}$
So

Now let's find their sum

$x + \frac{1}{x} = 2 + \sqrt{3} + 2 - \sqrt{3} = 4$
Now,
Let's find their product

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

We will cube the sum which we found above

So,

${(x + \frac{1}{x} )}^{3} = {(4)}^{3} \\ \\ {x}^{3} + \frac{1}{ {x}^{3} } + 3 \times x \times \frac{1}{x} (x + \frac{1}{x} ) = 64 \\ \\ {x}^{3} + \frac{1}{ {x}^{3} } + 3 \times 4 = 64 \\ \\ {x}^{3} + \frac{1}{ {x}^{3} } = 64 - 12 = 52$
Hence,

The answer is 52

The identity we used in the above answer is

(a + b) ³ =a³+b³+3ab( a + b)


Hence,
We got the final answer as 52.