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受欢迎的代数 >

化简 (8+6i)^4

解答

化简

(8 + 6 i)^{4}

解答

- 8432 + 5376 i

求解步骤

(8 + 6 i)^{4}

使用二项式定理:

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

a = 8, b = 6 i

= i = 0 \sum 4 (i 4) \cdot 8^{(4 - i)} (6 i)^{i}

展开求和

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

化简

\frac{4 !}{0 ! ( 4 - 0 ) !} \cdot 8^{4} (6 i)^{0} : 4096

化简

\frac{4 !}{1 ! ( 4 - 1 ) !} \cdot 8^{3} (6 i)^{1} : 12288 i

化简

\frac{4 !}{2 ! ( 4 - 2 ) !} \cdot 8^{2} (6 i)^{2} : 13824 i^{2}

化简

\frac{4 !}{3 ! ( 4 - 3 ) !} \cdot 8^{1} (6 i)^{3} : 6912 i^{3}

化简

\frac{4 !}{4 ! ( 4 - 4 ) !} \cdot 8^{0} (6 i)^{4} : 1296 i^{4}

= 4096 + 12288 i + 13824 i^{2} + 6912 i^{3} + 1296 i^{4}

化简

4096 + 12288 i + 13824 i^{2} + 6912 i^{3} + 1296 i^{4} : 5376 i - 8432

= 5376 i - 8432

Rewrite in standard complex form:

- 8432 + 5376 i

= - 8432 + 5376 i

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