What is (3+4i)^{10} ?

The solution to (3+4i)^{10} is -9653287+1476984i

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simplify (3+4i)^{10}

simplify

(3 + 4 i)^{10}

- 9653287 + 1476984 i

Solution steps

(3 + 4 i)^{10}

Apply binomial theorem:

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

a = 3, b = 4 i

= i = 0 \sum 10 (i 10) \cdot 3^{(10 - i)} (4 i)^{i}

Expand summation

= \frac{10 !}{0 ! ( 10 - 0 ) !} \cdot 3^{10} (4 i)^{0} + \frac{10 !}{1 ! ( 10 - 1 ) !} \cdot 3^{9} (4 i)^{1} + \frac{10 !}{2 ! ( 10 - 2 ) !} \cdot 3^{8} (4 i)^{2} + \frac{10 !}{3 ! ( 10 - 3 ) !} \cdot 3^{7} (4 i)^{3} + \frac{10 !}{4 ! ( 10 - 4 ) !} \cdot 3^{6} (4 i)^{4} + \frac{10 !}{5 ! ( 10 - 5 ) !} \cdot 3^{5} (4 i)^{5} + \frac{10 !}{6 ! ( 10 - 6 ) !} \cdot 3^{4} (4 i)^{6} + \frac{10 !}{7 ! ( 10 - 7 ) !} \cdot 3^{3} (4 i)^{7} + \frac{10 !}{8 ! ( 10 - 8 ) !} \cdot 3^{2} (4 i)^{8} + \frac{10 !}{9 ! ( 10 - 9 ) !} \cdot 3^{1} (4 i)^{9} + \frac{10 !}{10 ! ( 10 - 10 ) !} \cdot 3^{0} (4 i)^{10}

Simplify

\frac{10 !}{0 ! ( 10 - 0 ) !} \cdot 3^{10} (4 i)^{0} : 59049

Simplify

\frac{10 !}{1 ! ( 10 - 1 ) !} \cdot 3^{9} (4 i)^{1} : 787320 i

Simplify

\frac{10 !}{2 ! ( 10 - 2 ) !} \cdot 3^{8} (4 i)^{2} : 4723920 i^{2}

Simplify

\frac{10 !}{3 ! ( 10 - 3 ) !} \cdot 3^{7} (4 i)^{3} : 16796160 i^{3}

Simplify

\frac{10 !}{4 ! ( 10 - 4 ) !} \cdot 3^{6} (4 i)^{4} : 39191040 i^{4}

Simplify

\frac{10 !}{5 ! ( 10 - 5 ) !} \cdot 3^{5} (4 i)^{5} : 62705664 i^{5}

Simplify

\frac{10 !}{6 ! ( 10 - 6 ) !} \cdot 3^{4} (4 i)^{6} : 69672960 i^{6}

Simplify

\frac{10 !}{7 ! ( 10 - 7 ) !} \cdot 3^{3} (4 i)^{7} : 53084160 i^{7}

Simplify

\frac{10 !}{8 ! ( 10 - 8 ) !} \cdot 3^{2} (4 i)^{8} : 26542080 i^{8}

Simplify

\frac{10 !}{9 ! ( 10 - 9 ) !} \cdot 3^{1} (4 i)^{9} : 7864320 i^{9}

Simplify

\frac{10 !}{10 ! ( 10 - 10 ) !} \cdot 3^{0} (4 i)^{10} : 1048576 i^{10}

= 59049 + 787320 i + 4723920 i^{2} + 16796160 i^{3} + 39191040 i^{4} + 62705664 i^{5} + 69672960 i^{6} + 53084160 i^{7} + 26542080 i^{8} + 7864320 i^{9} + 1048576 i^{10}

Simplify

59049 + 787320 i + 4723920 i^{2} + 16796160 i^{3} + 39191040 i^{4} + 62705664 i^{5} + 69672960 i^{6} + 53084160 i^{7} + 26542080 i^{8} + 7864320 i^{9} + 1048576 i^{10} : 1476984 i - 9653287

= 1476984 i - 9653287

Rewrite in standard complex form:

- 9653287 + 1476984 i

= - 9653287 + 1476984 i

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