Question: \[ \text { 1. Express } \frac{(1+\sqrt{5})}{\left(2+\frac{5}{\sqrt{5}}\right)} \text { in the form } a+\sqrt{b} \text { where both } a \text { and } b \text { are integers. } \]
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To express \(\frac{(1+\sqrt{5})}{\left(2+\frac{5}{\sqrt{5}}\right)}\) in the form \(a+\sqrt{b}\), we first simplify the denominator: \[ 2 + \frac{5}{\sqrt{5}} = 2 + \sqrt{5} \cdot \frac{5}{5} = 2 + \sqrt{5} = 2 + \sqrt{5}. \] Now, substituting this back into our expression, we get: \[ \frac{(1+\sqrt{5})}{(2+\sqrt{5})}. \] Next, we can multiply the numerator and the denominator by the conjugate of the denominator to simplify further: \[ \frac{(1+\sqrt{5})(2-\sqrt{5})}{(2+\sqrt{5})(2-\sqrt{5})}. \] The denominator simplifies as follows: \[ (2+\sqrt{5})(2-\sqrt{5}) = 4 - 5 = -1. \] Now, simplifying the numerator: \[ (1+\sqrt{5})(2-\sqrt{5}) = 1 \cdot 2 + 1 \cdot (-\sqrt{5}) + \sqrt{5} \cdot 2 + \sqrt{5} \cdot (-\sqrt{5}) = 2 - \sqrt{5} + 2\sqrt{5} - 5 = -3 + \sqrt{5}. \] Putting it all together, we have: \[ \frac{-3 + \sqrt{5}}{-1} = 3 - \sqrt{5}. \] Thus, we can express our result as: \[ 3 - \sqrt{5}, \] identifying \(a = 3\) and \(b = 5\).
