展开 ((a+bi)^6)/((a-bi)^2)

解答

展开

\frac{( a + bi ) ^{6}}{( a - bi ) ^{2}}

\frac{a ^{8} + b ^{8} - 28 a ^{2} b ^{6} - 28 a ^{6} b ^{2} + 70 a ^{4} b ^{4}}{a ^{4} + b ^{4} + 2 a ^{2} b ^{2}} + i \frac{56 a ^{3} b ^{5} - 56 a ^{5} b ^{3} + 8 a ^{7} b - 8 a b ^{7}}{a ^{4} + b ^{4} + 2 a ^{2} b ^{2}}

求解步骤

\frac{( a + bi ) ^{6}}{( a - bi ) ^{2}}

(a - bi)^{2} = a^{2} - 2 iab - b^{2}

= \frac{( a + ib ) ^{6}}{a ^{2} - 2 iab - b ^{2}}

(a + bi)^{6} = a^{6} + 6 i a^{5} b - 15 a^{4} b^{2} - 20 i a^{3} b^{3} + 15 a^{2} b^{4} + 6 ia b^{5} - b^{6}

= \frac{a ^{6} + 6 i a ^{5} b - 15 a ^{4} b ^{2} - 20 i a ^{3} b ^{3} + 15 a ^{2} b ^{4} + 6 ia b ^{5} - b ^{6}}{a ^{2} - 2 iab - b ^{2}}

\frac{a ^{6} + 6 i a ^{5} b - 15 a ^{4} b ^{2} - 20 i a ^{3} b ^{3} + 15 a ^{2} b ^{4} + 6 ia b ^{5} - b ^{6}}{a ^{2} - 2 iab - b ^{2}}

有理化:

\frac{a ^{8} - 28 a ^{6} b ^{2} + 70 a ^{4} b ^{4} - 28 a ^{2} b ^{6} + i ( 8 a ^{7} b - 56 a ^{5} b ^{3} + 56 a ^{3} b ^{5} - 8 a b ^{7} ) + b ^{8}}{a ^{4} + 2 a ^{2} b ^{2} + b ^{4}}

= \frac{a ^{8} - 28 a ^{6} b ^{2} + 70 a ^{4} b ^{4} - 28 a ^{2} b ^{6} + b ^{8} + i ( 8 a ^{7} b - 56 a ^{5} b ^{3} + 56 a ^{3} b ^{5} - 8 a b ^{7} )}{a ^{4} + 2 a ^{2} b ^{2} + b ^{4}}

将

\frac{a ^{8} - 28 a ^{6} b ^{2} + 70 a ^{4} b ^{4} - 28 a ^{2} b ^{6} + b ^{8} + i ( 8 a ^{7} b - 56 a ^{5} b ^{3} + 56 a ^{3} b ^{5} - 8 a b ^{7} )}{a ^{4} + 2 a ^{2} b ^{2} + b ^{4}}

改写成标准复数形式:

\frac{a ^{8} - 28 a ^{6} b ^{2} + 70 a ^{4} b ^{4} - 28 a ^{2} b ^{6} + b ^{8}}{a ^{4} + 2 a ^{2} b ^{2} + b ^{4}} + \frac{8 a ^{7} b - 56 a ^{5} b ^{3} + 56 a ^{3} b ^{5} - 8 a b ^{7}}{a ^{4} + 2 a ^{2} b ^{2} + b ^{4}} i

= \frac{a ^{8} - 28 a ^{6} b ^{2} + 70 a ^{4} b ^{4} - 28 a ^{2} b ^{6} + b ^{8}}{a ^{4} + 2 a ^{2} b ^{2} + b ^{4}} + \frac{8 a ^{7} b - 56 a ^{5} b ^{3} + 56 a ^{3} b ^{5} - 8 a b ^{7}}{a ^{4} + 2 a ^{2} b ^{2} + b ^{4}} i

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