What is (3+4i)^{12} ?

The solution to (3+4i)^{12} is 32125393-242017776i

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simplify (3+4i)^{12}

simplify

(3 + 4 i)^{12}

32125393 - 242017776 i

Solution steps

(3 + 4 i)^{12}

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 12 (i 12) \cdot 3^{(12 - i)} (4 i)^{i}

Expand summation

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

Simplify

\frac{12 !}{0 ! ( 12 - 0 ) !} \cdot 3^{12} (4 i)^{0} : 531441

Simplify

\frac{12 !}{1 ! ( 12 - 1 ) !} \cdot 3^{11} (4 i)^{1} : 8503056 i

Simplify

\frac{12 !}{2 ! ( 12 - 2 ) !} \cdot 3^{10} (4 i)^{2} : 62355744 i^{2}

Simplify

\frac{12 !}{3 ! ( 12 - 3 ) !} \cdot 3^{9} (4 i)^{3} : 277136640 i^{3}

Simplify

\frac{12 !}{4 ! ( 12 - 4 ) !} \cdot 3^{8} (4 i)^{4} : 831409920 i^{4}

Simplify

\frac{12 !}{5 ! ( 12 - 5 ) !} \cdot 3^{7} (4 i)^{5} : 1773674496 i^{5}

Simplify

\frac{12 !}{6 ! ( 12 - 6 ) !} \cdot 3^{6} (4 i)^{6} : 2759049216 i^{6}

Simplify

\frac{12 !}{7 ! ( 12 - 7 ) !} \cdot 3^{5} (4 i)^{7} : 3153199104 i^{7}

Simplify

\frac{12 !}{8 ! ( 12 - 8 ) !} \cdot 3^{4} (4 i)^{8} : 2627665920 i^{8}

Simplify

\frac{12 !}{9 ! ( 12 - 9 ) !} \cdot 3^{3} (4 i)^{9} : 1557135360 i^{9}

Simplify

\frac{12 !}{10 ! ( 12 - 10 ) !} \cdot 3^{2} (4 i)^{10} : 622854144 i^{10}

Simplify

\frac{12 !}{11 ! ( 12 - 11 ) !} \cdot 3^{1} (4 i)^{11} : 150994944 i^{11}

Simplify

\frac{12 !}{12 ! ( 12 - 12 ) !} \cdot 3^{0} (4 i)^{12} : 16777216 i^{12}

= 531441 + 8503056 i + 62355744 i^{2} + 277136640 i^{3} + 831409920 i^{4} + 1773674496 i^{5} + 2759049216 i^{6} + 3153199104 i^{7} + 2627665920 i^{8} + 1557135360 i^{9} + 622854144 i^{10} + 150994944 i^{11} + 16777216 i^{12}

Simplify

531441 + 8503056 i + 62355744 i^{2} + 277136640 i^{3} + 831409920 i^{4} + 1773674496 i^{5} + 2759049216 i^{6} + 3153199104 i^{7} + 2627665920 i^{8} + 1557135360 i^{9} + 622854144 i^{10} + 150994944 i^{11} + 16777216 i^{12} : - 242017776 i + 32125393

= - 242017776 i + 32125393

Rewrite in standard complex form:

32125393 - 242017776 i

= 32125393 - 242017776 i

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