Опростете израза:

[Гост]

Опростете израза:
$$(\frac{a - \sqrt[3]{a}}{\sqrt[3]{a} - 1} - \frac{a + \sqrt[3]{a^{2}}}{\sqrt[3]{a} + 1}) : (\frac{\sqrt[4]{a} + \sqrt[4]{a^{3}}}{\sqrt{a} + 1} + \frac{1 - \sqrt{a}}{\sqrt[4]{a}})$$

[Davids]

Опростете израза:
$$(\frac{a - \sqrt[3]{a}}{\sqrt[3]{a} - 1} - \frac{a + \sqrt[3]{a^{2}}}{\sqrt[3]{a} + 1}) : (\frac{\sqrt[4]{a} + \sqrt[4]{a^{3}}}{\sqrt{a} + 1} + \frac{1 - \sqrt{a}}{\sqrt[4]{a}})$$

$$= (\frac{\sqrt[3]{a}(\sqrt[3]{a^2} - 1)}{\sqrt[3]{a} - 1} - \frac{\sqrt[3]{a^2}(\cancel{\sqrt[3]{a} + 1})}{\cancel{\sqrt[3]{a} + 1}}):(\frac{\sqrt[4]{a}(\cancel{1 + \sqrt{a}})}{\cancel{\sqrt{a} + 1}} + \frac{1 - \sqrt{a}}{\sqrt[4]{a}}) = $$
$$= (\frac{\sqrt[3]{a}(\sqrt[3]{a} + 1)(\cancel{\sqrt[3]{a} - 1})}{\cancel{\sqrt[3]{a} - 1}} - \sqrt[3]{a^2}):(\sqrt[4]{a} + \frac{1 - \sqrt{a}}{\sqrt[4]{a}}) =$$
$$= (\cancel{\sqrt[3]{a^2}} + 1 - \cancel{\sqrt[3]{a^2}}):(\frac{\cancel{\sqrt[4]{a^2}} + 1 - \cancel{\sqrt{a}}}{\sqrt[4]{a}}) =$$
$$= \boxed{\sqrt[4]{a}}$$

[S.B.]

Опростете израза:
$$(\frac{a - \sqrt[3]{a}}{\sqrt[3]{a} - 1} - \frac{a + \sqrt[3]{a^{2}}}{\sqrt[3]{a} + 1}) : (\frac{\sqrt[4]{a} + \sqrt[4]{a^{3}}}{\sqrt{a} + 1} + \frac{1 - \sqrt{a}}{\sqrt[4]{a}})$$

Още един поглед върху задачата...
[tex](\frac{a - \sqrt[3]{a}}{\sqrt[3]{a} - 1} - \frac{a + \sqrt[3]{a^{2}}}{\sqrt[3]{a} + 1}) : (\frac{\sqrt[4]{a} + \sqrt[4]{a^{3}}}{\sqrt{a} + 1} + \frac{1 - \sqrt{a}}{\sqrt[4]{a}} ) =[/tex]

[tex]= ( \frac{a - a^{\frac{1}{3}}}{a^{\frac{1}{3}} - 1} - \frac{a + a^{\frac{2}{3}}}{a^{\frac{1}{3}} + 1} ) : (\frac{a^{\frac{1}{4}} + a^{\frac{3}{4}}}{a^{\frac{1}{2}} + 1} + \frac{1 - a^{\frac{1}{2}}}{a^{\frac{1}{4}}}) =[/tex]

[tex]= (\frac{a - a^{\frac{1}{3}}}{a^{\frac{1}{3}} - 1} - \frac{a^{\frac{2}{3}}(a^{\frac{1}{3}} + 1)}{a^{\frac{1}{3}} + 1}) : (\frac{a^{\frac{1}{4}}(a^{\frac{1}{2}} + 1)}{a^{\frac{1}{2}} + 1} + \frac{1 - a^{\frac{1}{2}}}{a^{\frac{1}{4}}}) =[/tex]

[tex]= (\frac{a - a^{\frac{1}{3}}}{a^{\frac{1}{3}} - 1} - a^{\frac{2}{3}}) : ( a^{\frac{1}{4}} + \frac{1 - a^{\frac{1}{2}}}{a^{\frac{1}{4}}}) =[/tex]

[tex]=( \frac{a - a^{\frac{1}{3}} - a + a^{\frac{2}{3}}}{a^{\frac{1}{3}} - 1}) : ( \frac{a^{\frac{1}{2}} + 1 - a^{\frac{1}{2}}}{a^{\frac{1}{4}}}) =[/tex]

[tex]= (\frac{a^{\frac{2}{3}} - a^{\frac{1}{3}}}{a^{\frac{1}{3}} - 1}) : (\frac{1}{a^{\frac{1}{4}}}) =[/tex]

[tex]=( \frac{a^{\frac{1}{3}}(a^{\frac{1}{3}} - 1)}{a^{\frac{1}{3}} - 1}) . a^{\frac{1}{4}} = a^{\frac{1}{3}} . a^{\frac{1}{4}} = a^{\frac{1}{3} + \frac{1}{4}} = a^{\frac{7}{12}}[/tex]
$$ = \sqrt[12]{a^{7}}$$

[Davids]

Даа, видях си пропуска! Пропуснал съм да умножа и по едно в левите скоби, отговорът излиза точно $\sqrt[12]{a^7}$