Cancel out $\frac{a\left(a-b^{\frac12}\right)}{a+a^{\frac12}\cdot b^{\frac14}}$
Cancel out $$\dfrac{a\left(a-b^{\frac12}\right)}{a+a^{\frac12}\cdot b^{\frac14}}$$ The given expression is equal to $$\dfrac{a\left(a-b^{\frac12}\right)}{\left(a^\frac12\right)^2+a^\frac12\cdot b^\frac14}=\dfrac{a\left(a-b^{\frac12}\right)}{a^\frac12\left(a^\frac12+b^\frac14\right)}$$
$$a-b^{\frac{1}{2}} = \left( a^{\frac{1}{2}} - b^{\frac{1}{4}} \right)\left( a^{\frac{1}{2}} + b^{\frac{1}{4}} \right)$$