a) Ta có:
\(ab+b\sqrt{a}+\sqrt{a}+1=(ab+b\sqrt{a})+(\sqrt{a}+1)\)
\(=(ba+b\sqrt{a})+(\sqrt{a}+1)\)
\(=\left[ {b.\left( {\sqrt a .\sqrt a } \right) + b\sqrt a} \right] + \left( {\sqrt a + 1} \right)\)
\(=[(b\sqrt a).\sqrt a+ b\sqrt a.1]+(\sqrt a + 1)\)
\(=b\sqrt{a}(\sqrt{a}+1)+(\sqrt{a}+1)\)
\(=(\sqrt{a}+1)(b\sqrt{a}+1)\).
b) Ta có:
Cách 1: Sử dụng hằng đẳng thức số \(7\):
\(\sqrt{x^{3}}-\sqrt{y^{3}}+\sqrt{x^{2}y}-\sqrt{xy^{2}}\)
\(=[(\sqrt x)^3-(\sqrt y)^3]+ (\sqrt{x.xy}-\sqrt{y.xy})\)
\(=(\sqrt x-\sqrt y).[(\sqrt x)^2 + \sqrt x.\sqrt y+(\sqrt y)^2]\)
\(+ (\sqrt{x}.\sqrt{xy}-\sqrt{y}.\sqrt{xy})\)
\(=(\sqrt x-\sqrt y).[(\sqrt x)^2 + \sqrt x.\sqrt y+(\sqrt y)^2]\)
\(+ \sqrt{xy}.(\sqrt{x}-\sqrt{y})\)
\(=(\sqrt x-\sqrt y).[(\sqrt x)^2 + \sqrt x.\sqrt y+(\sqrt y)^2+\sqrt{xy}]\)
\(=(\sqrt x-\sqrt y).[(\sqrt x)^2 + 2\sqrt x.\sqrt y+(\sqrt y)^2]\)
\(=(\sqrt x-\sqrt y).(\sqrt x+\sqrt y)^2\).
Cách 2: Nhóm các hạng tử:
\(\sqrt{x^{3}}-\sqrt{y^{3}}+\sqrt{x^{2}y}-\sqrt{xy^{2}}\)
\(=x\sqrt{x}-y\sqrt{y}+x\sqrt{y}-y\sqrt{x}\)
\(=(x\sqrt{x}+x\sqrt{y})-(y\sqrt{x}+y\sqrt{y})\)
\(=x(\sqrt{x}+\sqrt{y})-y(\sqrt{y}+\sqrt{x})\)
\(=(\sqrt{x}+\sqrt{y})(x-y)\)
\(=(\sqrt{x}+\sqrt{y})(\sqrt x+\sqrt y)(\sqrt x -\sqrt y)\)
\(=(\sqrt{x}+\sqrt{y})^2(\sqrt{x}-\sqrt{y})\).
