Cho a và b là số dương, a ≠ b. Rút gọn biểu thức sau:

$A=\left[\frac{a-b}{a^{\frac{3}{4}}+a^{\frac{1}{2}}{b}^{\frac{1}{4}}}-\frac{a^{\frac{1}{2}}-b^{\frac{1}{2}}}{a^{\frac{1}{4}}+b^{\frac{1}{4}}}\right]:\left(a^{\frac{1}{4}}-b^{\frac{1}{4}}\right)$.

Đặt $B=\frac{a-b}{a^{\frac{3}{4}}+a^{\frac{1}{2}}{b}^{\frac{1}{4}}}-\frac{a^{\frac{1}{2}}-b^{\frac{1}{2}}}{a^{\frac{1}{4}}+b^{\frac{1}{4}}}=\frac{a-b}{a^{\frac{1}{2}}\left(a^{\frac{1}{4}}+b^{\frac{1}{4}}\right)}-\frac{a^{\frac{1}{2}}-b^{\frac{1}{2}}}{a^{\frac{1}{4}}+b^{\frac{1}{4}}}$

$=\frac{a-b-a^{\frac{1}{2}}\left(a^{\frac{1}{2}}-b^{\frac{1}{2}}\right)}{a^{\frac{1}{2}}\left(a^{\frac{1}{4}}+b^{\frac{1}{4}}\right)}$$=\frac{a-b-a+a^{\frac{1}{2}}{b}^{\frac{1}{2}}}{a^{\frac{1}{2}}\left(a^{\frac{1}{4}}+b^{\frac{1}{4}}\right)}$$=\frac{-b+a^{\frac{1}{2}}{b}^{\frac{1}{2}}}{a^{\frac{1}{2}}\left(a^{\frac{1}{4}}+b^{\frac{1}{4}}\right)}=\frac{b^{\frac{1}{2}}\left(a^{\frac{1}{2}}-b^{\frac{1}{2}}\right)}{a^{\frac{1}{2}}\left(a^{\frac{1}{4}}+b^{\frac{1}{4}}\right)}$

$=\frac{b^{\frac{1}{2}}\left(a^{\frac{1}{4}}-b^{\frac{1}{4}}\right)\left(a^{\frac{1}{4}}+b^{\frac{1}{4}}\right)}{a^{\frac{1}{2}}\left(a^{\frac{1}{4}}+b^{\frac{1}{4}}\right)}={\left(\frac{b}{a}\right)}^{\frac{1}{2}}\left(a^{\frac{1}{4}}-b^{\frac{1}{4}}\right)$

Do đó $A=\left[\frac{a-b}{a^{\frac{3}{4}}+a^{\frac{1}{2}}{b}^{\frac{1}{4}}}-\frac{a^{\frac{1}{2}}-b^{\frac{1}{2}}}{a^{\frac{1}{4}}+b^{\frac{1}{4}}}\right]:\left(a^{\frac{1}{4}}-b^{\frac{1}{4}}\right)$ $=\left[{\left(\frac{b}{a}\right)}^{\frac{1}{2}}\left(a^{\frac{1}{4}}-b^{\frac{1}{4}}\right)\right]:\left(a^{\frac{1}{4}}-b^{\frac{1}{4}}\right)$

$={\left(\frac{b}{a}\right)}^{\frac{1}{2}}\left(a^{\frac{1}{4}}-b^{\frac{1}{4}}\right)\cdot \frac{1}{a^{\frac{1}{4}}-b^{\frac{1}{4}}}={\left(\frac{b}{a}\right)}^{\frac{1}{2}}$.