What is (3+4i)^{14} ?

The solution to (3+4i)^{14} is 5583548873+2465133864i

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simplify (3+4i)^{14}

simplify

(3 + 4 i)^{14}

5583548873 + 2465133864 i

Solution steps

(3 + 4 i)^{14}

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 14 (i 14) \cdot 3^{(14 - i)} (4 i)^{i}

Expand summation

= \frac{14 !}{0 ! ( 14 - 0 ) !} \cdot 3^{14} (4 i)^{0} + \frac{14 !}{1 ! ( 14 - 1 ) !} \cdot 3^{13} (4 i)^{1} + \frac{14 !}{2 ! ( 14 - 2 ) !} \cdot 3^{12} (4 i)^{2} + \frac{14 !}{3 ! ( 14 - 3 ) !} \cdot 3^{11} (4 i)^{3} + \frac{14 !}{4 ! ( 14 - 4 ) !} \cdot 3^{10} (4 i)^{4} + \frac{14 !}{5 ! ( 14 - 5 ) !} \cdot 3^{9} (4 i)^{5} + \frac{14 !}{6 ! ( 14 - 6 ) !} \cdot 3^{8} (4 i)^{6} + \frac{14 !}{7 ! ( 14 - 7 ) !} \cdot 3^{7} (4 i)^{7} + \frac{14 !}{8 ! ( 14 - 8 ) !} \cdot 3^{6} (4 i)^{8} + \frac{14 !}{9 ! ( 14 - 9 ) !} \cdot 3^{5} (4 i)^{9} + \frac{14 !}{10 ! ( 14 - 10 ) !} \cdot 3^{4} (4 i)^{10} + \frac{14 !}{11 ! ( 14 - 11 ) !} \cdot 3^{3} (4 i)^{11} + \frac{14 !}{12 ! ( 14 - 12 ) !} \cdot 3^{2} (4 i)^{12} + \frac{14 !}{13 ! ( 14 - 13 ) !} \cdot 3^{1} (4 i)^{13} + \frac{14 !}{14 ! ( 14 - 14 ) !} \cdot 3^{0} (4 i)^{14}

Simplify

\frac{14 !}{0 ! ( 14 - 0 ) !} \cdot 3^{14} (4 i)^{0} : 4782969

Simplify

\frac{14 !}{1 ! ( 14 - 1 ) !} \cdot 3^{13} (4 i)^{1} : 89282088 i

Simplify

\frac{14 !}{2 ! ( 14 - 2 ) !} \cdot 3^{12} (4 i)^{2} : 773778096 i^{2}

Simplify

\frac{14 !}{3 ! ( 14 - 3 ) !} \cdot 3^{11} (4 i)^{3} : 4126816512 i^{3}

Simplify

\frac{14 !}{4 ! ( 14 - 4 ) !} \cdot 3^{10} (4 i)^{4} : 15131660544 i^{4}

Simplify

\frac{14 !}{5 ! ( 14 - 5 ) !} \cdot 3^{9} (4 i)^{5} : 40351094784 i^{5}

Simplify

\frac{14 !}{6 ! ( 14 - 6 ) !} \cdot 3^{8} (4 i)^{6} : 80702189568 i^{6}

Simplify

\frac{14 !}{7 ! ( 14 - 7 ) !} \cdot 3^{7} (4 i)^{7} : 122974765056 i^{7}

Simplify

\frac{14 !}{8 ! ( 14 - 8 ) !} \cdot 3^{6} (4 i)^{8} : 143470559232 i^{8}

Simplify

\frac{14 !}{9 ! ( 14 - 9 ) !} \cdot 3^{5} (4 i)^{9} : 127529385984 i^{9}

Simplify

\frac{14 !}{10 ! ( 14 - 10 ) !} \cdot 3^{4} (4 i)^{10} : 85019590656 i^{10}

Simplify

\frac{14 !}{11 ! ( 14 - 11 ) !} \cdot 3^{3} (4 i)^{11} : 41221619712 i^{11}

Simplify

\frac{14 !}{12 ! ( 14 - 12 ) !} \cdot 3^{2} (4 i)^{12} : 13740539904 i^{12}

Simplify

\frac{14 !}{13 ! ( 14 - 13 ) !} \cdot 3^{1} (4 i)^{13} : 2818572288 i^{13}

Simplify

\frac{14 !}{14 ! ( 14 - 14 ) !} \cdot 3^{0} (4 i)^{14} : 268435456 i^{14}

= 4782969 + 89282088 i + 773778096 i^{2} + 4126816512 i^{3} + 15131660544 i^{4} + 40351094784 i^{5} + 80702189568 i^{6} + 122974765056 i^{7} + 143470559232 i^{8} + 127529385984 i^{9} + 85019590656 i^{10} + 41221619712 i^{11} + 13740539904 i^{12} + 2818572288 i^{13} + 268435456 i^{14}

Simplify

4782969 + 89282088 i + 773778096 i^{2} + 4126816512 i^{3} + 15131660544 i^{4} + 40351094784 i^{5} + 80702189568 i^{6} + 122974765056 i^{7} + 143470559232 i^{8} + 127529385984 i^{9} + 85019590656 i^{10} + 41221619712 i^{11} + 13740539904 i^{12} + 2818572288 i^{13} + 268435456 i^{14} : 2465133864 i + 5583548873

= 2465133864 i + 5583548873

Rewrite in standard complex form:

5583548873 + 2465133864 i

= 5583548873 + 2465133864 i

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