展开-(5(2-x^{1/5})^4)/4

解答

展开

- \frac{5 ( 2 - x ^{\frac{1}{5}} ) ^{4}}{4}

- 20 + 40 x^{\frac{1}{5}} - 30 x^{\frac{2}{5}} + 10 x^{\frac{3}{5}} - \frac{5 x ^{\frac{4}{5}}}{4}

求解步骤

- \frac{5 ( 2 - x ^{\frac{1}{5}} ) ^{4}}{4}

(2 - x^{\frac{1}{5}})^{4} = 16 - 32 x^{\frac{1}{5}} + 24 x^{\frac{2}{5}} - 8 x^{\frac{3}{5}} + x^{\frac{4}{5}}

= - \frac{5 ( 24 x ^{\frac{2}{5}} + x ^{\frac{4}{5}} - 32 x ^{\frac{1}{5}} - 8 x ^{\frac{3}{5}} + 16 )}{4}

乘开

5 (16 - 32 x^{\frac{1}{5}} + 24 x^{\frac{2}{5}} - 8 x^{\frac{3}{5}} + x^{\frac{4}{5}}) : 80 - 160 x^{\frac{1}{5}} + 120 x^{\frac{2}{5}} - 40 x^{\frac{3}{5}} + 5 x^{\frac{4}{5}}

= - \frac{120 x ^{\frac{2}{5}} + 5 x ^{\frac{4}{5}} - 160 x ^{\frac{1}{5}} - 40 x ^{\frac{3}{5}} + 80}{4}

使用分式法则:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{80 - 160 x ^{\frac{1}{5}} + 120 x ^{\frac{2}{5}} - 40 x ^{\frac{3}{5}} + 5 x ^{\frac{4}{5}}}{4} = - (\frac{80}{4}) - (- \frac{160 x ^{\frac{1}{5}}}{4}) - (\frac{120 x ^{\frac{2}{5}}}{4}) - (- \frac{40 x ^{\frac{3}{5}}}{4}) - (\frac{5 x ^{\frac{4}{5}}}{4})

= - (\frac{80}{4}) - (- \frac{160 x ^{\frac{1}{5}}}{4}) - (\frac{120 x ^{\frac{2}{5}}}{4}) - (- \frac{40 x ^{\frac{3}{5}}}{4}) - (\frac{5 x ^{\frac{4}{5}}}{4})

去除括号:

(a) = a, - (- a) = a

= - \frac{80}{4} + \frac{160 x ^{\frac{1}{5}}}{4} - \frac{120 x ^{\frac{2}{5}}}{4} + \frac{40 x ^{\frac{3}{5}}}{4} - \frac{5 x ^{\frac{4}{5}}}{4}

\frac{80}{4} = 20

\frac{160 x ^{\frac{1}{5}}}{4} = 40 x^{\frac{1}{5}}

\frac{120 x ^{\frac{2}{5}}}{4} = 30 x^{\frac{2}{5}}

\frac{40 x ^{\frac{3}{5}}}{4} = 10 x^{\frac{3}{5}}

= - 20 + 40 x^{\frac{1}{5}} - 30 x^{\frac{2}{5}} + 10 x^{\frac{3}{5}} - \frac{5 x ^{\frac{4}{5}}}{4}

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