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受欢迎的代数 >

因式分解 e^{20/13}-1

解答

因式分解

e^{\frac{20}{13}} - 1

解答

(e^{\frac{2}{13}} + 1) (e^{\frac{8}{13}} - e^{\frac{6}{13}} + e^{\frac{4}{13}} - e^{\frac{2}{13}} + 1) (13 e + 1) (e^{\frac{4}{13}} - e^{\frac{3}{13}} + e^{\frac{2}{13}} - 13 e + 1) (13 e - 1) (e^{\frac{4}{13}} + e^{\frac{3}{13}} + e^{\frac{2}{13}} + 13 e + 1)

求解步骤

e^{\frac{20}{13}} - 1

将

e^{\frac{20}{13}}

改写为

(13 e)^{20}

= (13 e)^{20} - 1

将

13 e^{20} - 1

改写为

((13 e)^{10})^{2} - 1^{2}

= ((13 e)^{10})^{2} - 1^{2}

使用平方差公式:

x^{2} - y^{2} = (x + y) (x - y)

((13 e)^{10})^{2} - 1^{2} = ((13 e)^{10} + 1) ((13 e)^{10} - 1)

= ((13 e)^{10} + 1) ((13 e)^{10} - 1)

分解

(13 e)^{10} + 1 : ((13 e)^{2} + 1) ((13 e)^{8} - (13 e)^{6} + (13 e)^{4} - (13 e)^{2} + 1)

= ((13 e)^{2} + 1) ((13 e)^{8} - (13 e)^{6} + (13 e)^{4} - (13 e)^{2} + 1) ((13 e)^{10} - 1)

分解

(13 e)^{10} - 1 : (13 e + 1) ((13 e)^{4} - (13 e)^{3} + (13 e)^{2} - 13 e + 1) (13 e - 1) ((13 e)^{4} + (13 e)^{3} + (13 e)^{2} + 13 e + 1)

= ((13 e)^{2} + 1) ((13 e)^{8} - (13 e)^{6} + (13 e)^{4} - (13 e)^{2} + 1) (13 e + 1) ((13 e)^{4} - (13 e)^{3} + (13 e)^{2} - 13 e + 1) (13 e - 1) ((13 e)^{4} + (13 e)^{3} + (13 e)^{2} + 13 e + 1)

(13 e)^{2} = e^{\frac{2}{13}}

= (e^{\frac{2}{13}} + 1) ((13 e)^{8} + (13 e)^{4} - (13 e)^{6} - (13 e)^{2} + 1) (13 e + 1) (- 13 e + (13 e)^{4} + (13 e)^{2} - (13 e)^{3} + 1) (13 e - 1) (13 e + (13 e)^{4} + (13 e)^{3} + (13 e)^{2} + 1)

(13 e)^{8} = e^{\frac{8}{13}}

(13 e)^{6} = e^{\frac{6}{13}}

(13 e)^{4} = e^{\frac{4}{13}}

(13 e)^{2} = e^{\frac{2}{13}}

= (e^{\frac{2}{13}} + 1) (e^{\frac{8}{13}} + e^{\frac{4}{13}} - e^{\frac{6}{13}} - e^{\frac{2}{13}} + 1) (13 e + 1) (- 13 e + (13 e)^{4} + (13 e)^{2} - (13 e)^{3} + 1) (13 e - 1) (13 e + (13 e)^{4} + (13 e)^{3} + (13 e)^{2} + 1)

(13 e)^{4} = e^{\frac{4}{13}}

(13 e)^{3} = e^{\frac{3}{13}}

(13 e)^{2} = e^{\frac{2}{13}}

= (e^{\frac{2}{13}} + 1) (e^{\frac{8}{13}} + e^{\frac{4}{13}} - e^{\frac{6}{13}} - e^{\frac{2}{13}} + 1) (13 e + 1) (e^{\frac{4}{13}} + e^{\frac{2}{13}} - e^{\frac{3}{13}} - 13 e + 1) (13 e - 1) (13 e + (13 e)^{4} + (13 e)^{3} + (13 e)^{2} + 1)

(13 e)^{4} = e^{\frac{4}{13}}

(13 e)^{3} = e^{\frac{3}{13}}

(13 e)^{2} = e^{\frac{2}{13}}

= (e^{\frac{2}{13}} + 1) (e^{\frac{8}{13}} + e^{\frac{4}{13}} - e^{\frac{6}{13}} - e^{\frac{2}{13}} + 1) (13 e + 1) (e^{\frac{4}{13}} + e^{\frac{2}{13}} - e^{\frac{3}{13}} - 13 e + 1) (13 e - 1) (e^{\frac{4}{13}} + e^{\frac{3}{13}} + e^{\frac{2}{13}} + 13 e + 1)

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