Question

If x + (1/x) = 3â2, then what is the value of x5 + (1/x5)?

Answer 1

Answer:

$-\frac{57}{32}$

Step-by-step explanation:

$x + \frac{1}{x} = \frac{3}{2}$

$(x + \frac{1}{x})^2 = \frac{9}{4} = x^2 + \frac{1}{x^2} + 2$

$x^2 + \frac{1}{x^2} = \frac{9}{4} - 2 = \frac{1}{4}$

$(x + \frac{1}{x})^3 = \frac{27}{8} = x^3 + \frac{1}{x^3} + 3(x + \frac{1}{x})$

$x^3 + \frac{1}{x^3} = \frac{27}{8} - 3(x + \frac{1}{x}) = \frac{27}{8} - 3(\frac{3}{2}) = -\frac{9}{8}$

Now, $(x^2 + \frac{1}{x^2})(x^3 + \frac{1}{x^3}) = (\frac{1}{4})(-\frac{9}{8}) = x^5 + \frac{1}{x^5} + x + \frac{1}{x}$

So, $x^5 + \frac{1}{x^5} = (\frac{1}{4})(-\frac{9}{8}) - (x + \frac{1}{x}) = -\frac{9}{32} - \frac{3}{2} = - \frac{57}{32}$