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受欢迎的代数 >

化简 (4+3i)^6

解答

化简

(4 + 3 i)^{6}

解答

- 11753 - 10296 i

求解步骤

(4 + 3 i)^{6}

使用二项式定理:

(a + b)^{n} = i = 0 \sum n (i n) a^{(n - i)} b^{i}

a = 4, b = 3 i

= i = 0 \sum 6 (i 6) \cdot 4^{(6 - i)} (3 i)^{i}

展开求和

= \frac{6 !}{0 ! ( 6 - 0 ) !} \cdot 4^{6} (3 i)^{0} + \frac{6 !}{1 ! ( 6 - 1 ) !} \cdot 4^{5} (3 i)^{1} + \frac{6 !}{2 ! ( 6 - 2 ) !} \cdot 4^{4} (3 i)^{2} + \frac{6 !}{3 ! ( 6 - 3 ) !} \cdot 4^{3} (3 i)^{3} + \frac{6 !}{4 ! ( 6 - 4 ) !} \cdot 4^{2} (3 i)^{4} + \frac{6 !}{5 ! ( 6 - 5 ) !} \cdot 4^{1} (3 i)^{5} + \frac{6 !}{6 ! ( 6 - 6 ) !} \cdot 4^{0} (3 i)^{6}

化简

\frac{6 !}{0 ! ( 6 - 0 ) !} \cdot 4^{6} (3 i)^{0} : 4096

化简

\frac{6 !}{1 ! ( 6 - 1 ) !} \cdot 4^{5} (3 i)^{1} : 18432 i

化简

\frac{6 !}{2 ! ( 6 - 2 ) !} \cdot 4^{4} (3 i)^{2} : 34560 i^{2}

化简

\frac{6 !}{3 ! ( 6 - 3 ) !} \cdot 4^{3} (3 i)^{3} : 34560 i^{3}

化简

\frac{6 !}{4 ! ( 6 - 4 ) !} \cdot 4^{2} (3 i)^{4} : 19440 i^{4}

化简

\frac{6 !}{5 ! ( 6 - 5 ) !} \cdot 4^{1} (3 i)^{5} : 5832 i^{5}

化简

\frac{6 !}{6 ! ( 6 - 6 ) !} \cdot 4^{0} (3 i)^{6} : 729 i^{6}

= 4096 + 18432 i + 34560 i^{2} + 34560 i^{3} + 19440 i^{4} + 5832 i^{5} + 729 i^{6}

化简

4096 + 18432 i + 34560 i^{2} + 34560 i^{3} + 19440 i^{4} + 5832 i^{5} + 729 i^{6} : - 10296 i - 11753

= - 10296 i - 11753

Rewrite in standard complex form:

- 11753 - 10296 i

= - 11753 - 10296 i

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