Giả thiết điều kiện trong căn đều có nghĩa
$1)5 + \sqrt 5 = {\sqrt 5 ^2} + \sqrt 5 = \sqrt 5 \left( {\sqrt 5 + 1} \right)$
$\begin{array}{l} 2)\sqrt {{x^2} - {y^2}} + \sqrt {x - y} = \sqrt {\left( {x - y} \right)\left( {x + y} \right)} + \sqrt {x - y} \\ = \sqrt {x - y} \left( {\sqrt {x + y} + 1} \right)\\ 3)x + y - 2\sqrt {xy} = {\sqrt x ^2} + {\sqrt y ^2} - 2\sqrt {xy} = {\left( {\sqrt x - \sqrt y } \right)^2} \end{array}$
4)Sửa đề: $\sqrt{ax}-\sqrt{ay}+\sqrt{bx}-\sqrt{by}$
$\begin{array}{l} \sqrt {ax} - \sqrt {ay} + \sqrt {bx} - \sqrt {by} \\ = \sqrt a \left( {\sqrt x - \sqrt y } \right) + \sqrt b \left( {\sqrt x - \sqrt y } \right)\\ = \left( {\sqrt x - \sqrt y } \right)\left( {\sqrt a + \sqrt b } \right) \end{array}$
5)
$\begin{array}{l} x + y + z + 2\sqrt {xy} \\ = {\left( {\sqrt x + \sqrt y } \right)^2} - \left( { - z} \right)\\ = \left( {\sqrt x + \sqrt y - \sqrt { - z} } \right)\left( {\sqrt x + \sqrt y + \sqrt { - z} } \right) \end{array}$
6)
$\begin{array}{l} x + 5\sqrt x + 6\\ = x + 2\sqrt x + 3\sqrt x + 6\\ = \sqrt x \left( {\sqrt x + 2} \right) + 3\left( {\sqrt x + 2} \right)\\ = \left( {\sqrt x + 2} \right)\left( {\sqrt x + 3} \right) \end{array}$
