\( x=4-\sqrt{15} \), find the value of \( \left(x+\frac{1}{x}\right) \)

To find the value of \( x + \frac{1}{x} \) where \( x = 4 - \sqrt{15} \), we first need to calculate \( \frac{1}{x} \). 1. Calculate \( \frac{1}{x} \): \[ \frac{1}{x} = \frac{1}{4 - \sqrt{15}} \] To rationalize the denominator, we multiply the numerator and the denominator by the conjugate \( 4 + \sqrt{15} \): \[ \frac{1}{x} = \frac{4 + \sqrt{15}}{(4 - \sqrt{15})(4 + \sqrt{15})} = \frac{4 + \sqrt{15}}{16 - 15} = 4 + \sqrt{15} \] 2. Now we compute \( x + \frac{1}{x} \): \[ x + \frac{1}{x} = (4 - \sqrt{15}) + (4 + \sqrt{15}) = 4 - \sqrt{15} + 4 + \sqrt{15} \] The \( -\sqrt{15} \) and \( \sqrt{15} \) cancel out: \[ x + \frac{1}{x} = 8 \] Thus, the value of \( x + \frac{1}{x} \) is \( \boxed{8} \).