sqrt(s+2 sqrt(6))+sqrt(8-2 sqrt(15))

To simplify the expression \( \sqrt{s+2 \sqrt{6}}+\sqrt{8-2 \sqrt{15}} \), we need to find values of \( s \) that make the expression simpler.<br /><br />First, let's consider the term \( \sqrt{8-2 \sqrt{15}} \). We can try to express it in the form \( \sqrt{a} - \sqrt{b} \).<br /><br />Assume:<br />\[ \sqrt{8-2\sqrt{15}} = \sqrt{a} - \sqrt{b} \]<br /><br />Squaring both sides:<br />\[ 8 - 2\sqrt{15} = a + b - 2\sqrt{ab} \]<br /><br />By comparing rational and irrational parts, we get:<br />\[ a + b = 8 \]<br />\[ -2\sqrt{15} = -2\sqrt{ab} \implies ab = 15 \]<br /><br />Solving these equations:<br />\[ a + b = 8 \]<br />\[ ab = 15 \]<br /><br />The solutions to these equations are the roots of the quadratic equation:<br />\[ t^2 - (a+b)t + ab = 0 \]<br />\[ t^2 - 8t + 15 = 0 \]<br /><br />Solving this quadratic equation:<br />\[ t = \frac{8 \pm \sqrt{64 - 60}}{2} = \frac{8 \pm 2}{2} \]<br />\[ t = 5 \text{ or } 3 \]<br /><br />Thus, \( a = 5 \) and \( b = 3 \). Therefore:<br />\[ \sqrt{8-2\sqrt{15}} = \sqrt{5} - \sqrt{3} \]<br /><br />Now, consider the term \( \sqrt{s + 2\sqrt{6}} \). We want to find \( s \) such that:<br />\[ \sqrt{s + 2\sqrt{6}} = \sqrt{a} + \sqrt{b} \]<br /><br />Squaring both sides:<br />\[ s + 2\sqrt{6} = a + b + 2\sqrt{ab} \]<br /><br />By comparing rational and irrational parts, we get:<br />\[ a + b = s \]<br />\[ 2\sqrt{6} = 2\sqrt{ab} \implies ab = 6 \]<br /><br />Solving these equations:<br />\[ a + b = s \]<br />\[ ab = 6 \]<br /><br />The solutions to these equations are the roots of the quadratic equation:<br />\[ t^2 - st + 6 = 0 \]<br /><br />We need to find \( s \) such that the roots are real numbers. Let's assume \( a = 3 \) and \( b = 2 \):<br />\[ s = a + b = 3 + 2 = 5 \]<br /><br />Thus:<br />\[ \sqrt{s + 2\sqrt{6}} = \sqrt{3} + \sqrt{2} \]<br /><br />Putting it all together:<br />\[ \sqrt{5 + 2\sqrt{6}} + \sqrt{8 - 2\sqrt{15}} = (\sqrt{3} + \sqrt{2}) + (\sqrt{5} - \sqrt{3}) \]<br />\[ = \sqrt{2} + \sqrt{5} \]<br /><br />Therefore, the simplified form of the given expression is:<br />\[ \boxed{\sqrt{2} + \sqrt{5}} \]