Question

If x=2+√3. Find x^3+1/x^3

Answer 1

Here is your answer:

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

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