化简 (a+ib)^3

解答

化简

(a + ib)^{3}

(a^{3} - 3 a b^{2}) + (3 a^{2} b - b^{3}) i

求解步骤

(a + ib)^{3}

使用完全立方公式:

(a + b)^{3} = a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}

(a + ib)^{3} = a^{3} + 3 a^{2} ib + 3 a (ib)^{2} + (ib)^{3}

= a^{3} + 3 a^{2} ib + 3 a (ib)^{2} + (ib)^{3}

3 a^{2} ib = 3 a^{2} bi

3 a (ib)^{2} = - 3 a b^{2}

(ib)^{3} = - i b^{3}

= a^{3} + 3 a^{2} bi - 3 a b^{2} - i b^{3}

Rewrite in standard complex form:

(a^{3} - 3 a b^{2}) + (3 a^{2} b - b^{3}) i

= (a^{3} - 3 a b^{2}) + (3 a^{2} b - b^{3}) i

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