Question

Please solve this surds

Answer 1

`√(8) =2√(2) \approx{2.8284...}`

Explanation:

`\sqrt{5+2√(6)} -\frac{1}{\sqrt{5+2√(6)} } \\`

1. put to the same denominator:

`=\frac{5+2√(6)-1 }{\sqrt{5+2√(6)} } \\=\frac{4+2√(6) }{\sqrt{5+2√(6)} }`

2. Make the denominator rationnel:

`\frac{4+2√(6) }{\sqrt{5+2√(6)} }\\\\\\=\frac{ (4+2√(6) )*(\sqrt{5-2√(6)} ) } {25-4*6} \\\\\\=(4+2√(6) )*(\sqrt{5-2√(6) } )`

3. expand

`(4+2√(6) )*(\sqrt{5-2√(6) } ) \\\\=\sqrt{(4+2√(6))^2 *(5-2√(6) )}\\\\=\sqrt{(16+4*6+16√(6))*( 5-2√(6) ) } \\\\=\sqrt{(40+16√(6))*(5-2√(6)) } \\\\=\sqrt{200+80√(6)-80√(6) -32*6 } \\\\=√(200-192) \\\\=√(8)`