$a){\left(a^{\frac{1}{2}}+b^{\frac{1}{2}}\right)}^2={\left(a^{\frac{1}{2}}\right)}^2+2\cdot a^{\frac{1}{2}}\cdot b^{\frac{1}{2}}+{\left(b^{\frac{1}{2}}\right)}^2=a+2\sqrt{ab}+b$  

 

$b){\left(x^{\frac{3}{2}}-y^{\frac{3}{2}}\right)}^2={\left(x^{\frac{3}{2}}\right)}^2-2\cdot x^{\frac{3}{2}}\cdot y^{\frac{3}{2}}+{\left(y^{\frac{3}{2}}\right)}^2=x^3-2\sqrt{x^3{y}^3}+y^3$ $b){\left(x^{\frac{3}{2}}-y^{\frac{3}{2}}\right)}^2={\left(x^{\frac{3}{2}}\right)}^2-2\cdot x^{\frac{3}{2}}\cdot y^{\frac{3}{2}}+{\left(y^{\frac{3}{2}}\right)}^2=x^3-2\sqrt{x^3{y}^3}+y^3$  

 

$c)\left(2x^{\frac{1}{2}}+y^{\frac{1}{4}}\right)\left(2x^{\frac{1}{2}}-y^{\frac{1}{4}}\right)={\left(2x^{\frac{1}{2}}\right)}^2-{\left(y^{\frac{1}{4}}\right)}^2=4x-\sqrt{y}$  

 

$d){\left(2\sqrt{a}+3\sqrt{b}\right)}^2={\left(2\sqrt{a}\right)}^2+2\cdot 2\sqrt{a}\cdot 3\sqrt{b}+{\left(3\sqrt{b}\right)}^2=4a+12\sqrt{ab}+9b$  

 

$e){\left(\frac{1}{2}\sqrt{x}-\frac{1}{3}\sqrt{y}\right)}^2={\left(\frac{1}{2}\sqrt{x}\right)}^2-2\cdot \frac{1}{2}\sqrt{x}\cdot \frac{1}{3}\sqrt{y}+{\left(\frac{1}{3}\sqrt{y}\right)}^2=\frac{x}{4}-\frac{\sqrt{xy}}{3}+\frac{y}{9}$  

 

$f)\left(2\sqrt{2x}-2\sqrt{3y}\right)\left(2\sqrt{2x}+2\sqrt{3y}\right)={\left(2\sqrt{2x}\right)}^2-{\left(2\sqrt{3y}\right)}^2=4\cdot 2x-4\cdot 3y=8x-12y$