ess \( \frac{(1+\sqrt{5})}{\left(2+\frac{5}{\sqrt{5}}\right)} \) in the form \( a+\sqrt{b} \) where both \( a \) and \( b \) are integ

To express \( \frac{(1+\sqrt{5})}{\left(2+\frac{5}{\sqrt{5}}\right)} \), we first simplify the denominator \( 2 + \frac{5}{\sqrt{5}} \). Calculating \( \frac{5}{\sqrt{5}} \) gives us \( \sqrt{5} \). So, the denominator simplifies to \( 2 + \sqrt{5} \). Now, we have: \[ \frac{(1+\sqrt{5})}{(2+\sqrt{5})} \] To simplify this fraction, we can multiply the numerator and the denominator by the conjugate of the denominator, \( 2 - \sqrt{5} \): \[ \frac{(1+\sqrt{5})(2-\sqrt{5})}{(2+\sqrt{5})(2-\sqrt{5})} \] Now we calculate the denominator: \[ (2+\sqrt{5})(2-\sqrt{5}) = 2^2 - (\sqrt{5})^2 = 4 - 5 = -1 \] Next, we calculate the numerator: \[ (1+\sqrt{5})(2-\sqrt{5}) = 1\cdot2 + 1\cdot(-\sqrt{5}) + \sqrt{5}\cdot2 + \sqrt{5}\cdot(-\sqrt{5}) = 2 - \sqrt{5} + 2\sqrt{5} - 5 = -3 + \sqrt{5} \] So, we combine these results: \[ \frac{-3+\sqrt{5}}{-1} = 3 - \sqrt{5} \] Thus, in the form \( a + \sqrt{b} \), we have: \[ 3 - \sqrt{5} = 3 + (-1)\sqrt{5} \] Hence, \( a = 3 \) and \( b = -5 \). However, since \( b \) must be positive, we can state our final result as: \[ 3 + \sqrt{5} \] where \( a = 3 \) and \( b = 5 \).