化简 (1-x^{16})^4(1-x)^{-4}

解答

化简

(1 - x^{16})^{4} (1 - x)^{- 4}

(x + 1)^{4} (x^{2} + 1)^{4} (x^{2} + 2 x + 1)^{4} (x^{2} - 2 x + 1)^{4} (x^{4} + 2 x^{2} + 1)^{4} (x^{4} - 2 x^{2} + 1)^{4}

求解步骤

(1 - x^{16})^{4} (1 - x)^{- 4}

使用指数法则:

a^{- b} = \frac{1}{a ^{b}}

= \frac{1}{( - x + 1 ) ^{4}} (- x^{16} + 1)^{4}

分式相乘:

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

= \frac{1 \cdot ( 1 - x ^{16} ) ^{4}}{( 1 - x ) ^{4}}

乘以:

1 \cdot (1 - x^{16})^{4} = (1 - x^{16})^{4}

= \frac{( 1 - x ^{16} ) ^{4}}{( 1 - x ) ^{4}}

分解

(- x^{16} + 1)^{4} : (x + 1)^{4} (x - 1)^{4} (x^{2} + 1)^{4} (x^{2} + 2 x + 1)^{4} (x^{2} - 2 x + 1)^{4} (x^{4} + 2 x^{2} + 1)^{4} (x^{4} - 2 x^{2} + 1)^{4}

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

消掉

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

= (x + 1)^{4} (x^{2} + 1)^{4} (x^{2} + 2 x + 1)^{4} (x^{2} - 2 x + 1)^{4} (x^{4} + 2 x^{2} + 1)^{4} (x^{4} - 2 x^{2} + 1)^{4}

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