Question

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Answer 1

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

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

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

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

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

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Answer 2

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

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

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

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

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

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