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受欢迎的代数 >

化简 (4+3i)^9

解答

化简

(4 + 3 i)^{9}

解答

1721764 - 922077 i

求解步骤

(4 + 3 i)^{9}

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 9 (i 9) \cdot 4^{(9 - i)} (3 i)^{i}

展开求和

= \frac{9 !}{0 ! ( 9 - 0 ) !} \cdot 4^{9} (3 i)^{0} + \frac{9 !}{1 ! ( 9 - 1 ) !} \cdot 4^{8} (3 i)^{1} + \frac{9 !}{2 ! ( 9 - 2 ) !} \cdot 4^{7} (3 i)^{2} + \frac{9 !}{3 ! ( 9 - 3 ) !} \cdot 4^{6} (3 i)^{3} + \frac{9 !}{4 ! ( 9 - 4 ) !} \cdot 4^{5} (3 i)^{4} + \frac{9 !}{5 ! ( 9 - 5 ) !} \cdot 4^{4} (3 i)^{5} + \frac{9 !}{6 ! ( 9 - 6 ) !} \cdot 4^{3} (3 i)^{6} + \frac{9 !}{7 ! ( 9 - 7 ) !} \cdot 4^{2} (3 i)^{7} + \frac{9 !}{8 ! ( 9 - 8 ) !} \cdot 4^{1} (3 i)^{8} + \frac{9 !}{9 ! ( 9 - 9 ) !} \cdot 4^{0} (3 i)^{9}

化简

\frac{9 !}{0 ! ( 9 - 0 ) !} \cdot 4^{9} (3 i)^{0} : 262144

化简

\frac{9 !}{1 ! ( 9 - 1 ) !} \cdot 4^{8} (3 i)^{1} : 1769472 i

化简

\frac{9 !}{2 ! ( 9 - 2 ) !} \cdot 4^{7} (3 i)^{2} : 5308416 i^{2}

化简

\frac{9 !}{3 ! ( 9 - 3 ) !} \cdot 4^{6} (3 i)^{3} : 9289728 i^{3}

化简

\frac{9 !}{4 ! ( 9 - 4 ) !} \cdot 4^{5} (3 i)^{4} : 10450944 i^{4}

化简

\frac{9 !}{5 ! ( 9 - 5 ) !} \cdot 4^{4} (3 i)^{5} : 7838208 i^{5}

化简

\frac{9 !}{6 ! ( 9 - 6 ) !} \cdot 4^{3} (3 i)^{6} : 3919104 i^{6}

化简

\frac{9 !}{7 ! ( 9 - 7 ) !} \cdot 4^{2} (3 i)^{7} : 1259712 i^{7}

化简

\frac{9 !}{8 ! ( 9 - 8 ) !} \cdot 4^{1} (3 i)^{8} : 236196 i^{8}

化简

\frac{9 !}{9 ! ( 9 - 9 ) !} \cdot 4^{0} (3 i)^{9} : 19683 i^{9}

= 262144 + 1769472 i + 5308416 i^{2} + 9289728 i^{3} + 10450944 i^{4} + 7838208 i^{5} + 3919104 i^{6} + 1259712 i^{7} + 236196 i^{8} + 19683 i^{9}

化简

262144 + 1769472 i + 5308416 i^{2} + 9289728 i^{3} + 10450944 i^{4} + 7838208 i^{5} + 3919104 i^{6} + 1259712 i^{7} + 236196 i^{8} + 19683 i^{9} : - 922077 i + 1721764

= - 922077 i + 1721764

Rewrite in standard complex form:

1721764 - 922077 i

= 1721764 - 922077 i

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