What is (1+sqrt(3)i)^{13} ?

The solution to (1+sqrt(3)i)^{13} is 4096+4096sqrt(3)i

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simplify (1+sqrt(3)i)^{13}

simplify

(1 + 3 i)^{13}

4096 + 40963 i

Solution steps

(1 + 3 i)^{13}

Apply binomial theorem:

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

a = 1, b = 3 i

= i = 0 \sum 13 (i 13) \cdot 1^{(13 - i)} (3 i)^{i}

Expand summation

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

Simplify

\frac{13 !}{0 ! ( 13 - 0 ) !} \cdot 1^{13} (3 i)^{0} : 1

Simplify

\frac{13 !}{1 ! ( 13 - 1 ) !} \cdot 1^{12} (3 i)^{1} : 133 i

Simplify

\frac{13 !}{2 ! ( 13 - 2 ) !} \cdot 1^{11} (3 i)^{2} : 234 i^{2}

Simplify

\frac{13 !}{3 ! ( 13 - 3 ) !} \cdot 1^{10} (3 i)^{3} : 8583 i^{3}

Simplify

\frac{13 !}{4 ! ( 13 - 4 ) !} \cdot 1^{9} (3 i)^{4} : 6435 i^{4}

Simplify

\frac{13 !}{5 ! ( 13 - 5 ) !} \cdot 1^{8} (3 i)^{5} : 115833 i^{5}

Simplify

\frac{13 !}{6 ! ( 13 - 6 ) !} \cdot 1^{7} (3 i)^{6} : 46332 i^{6}

Simplify

\frac{13 !}{7 ! ( 13 - 7 ) !} \cdot 1^{6} (3 i)^{7} : 463323 i^{7}

Simplify

\frac{13 !}{8 ! ( 13 - 8 ) !} \cdot 1^{5} (3 i)^{8} : 104247 i^{8}

Simplify

\frac{13 !}{9 ! ( 13 - 9 ) !} \cdot 1^{4} (3 i)^{9} : 579153 i^{9}

Simplify

\frac{13 !}{10 ! ( 13 - 10 ) !} \cdot 1^{3} (3 i)^{10} : 69498 i^{10}

Simplify

\frac{13 !}{11 ! ( 13 - 11 ) !} \cdot 1^{2} (3 i)^{11} : 189543 i^{11}

Simplify

\frac{13 !}{12 ! ( 13 - 12 ) !} \cdot 1^{1} (3 i)^{12} : 9477 i^{12}

Simplify

\frac{13 !}{13 ! ( 13 - 13 ) !} \cdot 1^{0} (3 i)^{13} : 7293 i^{13}

= 1 + 133 i + 234 i^{2} + 8583 i^{3} + 6435 i^{4} + 115833 i^{5} + 46332 i^{6} + 463323 i^{7} + 104247 i^{8} + 579153 i^{9} + 69498 i^{10} + 189543 i^{11} + 9477 i^{12} + 7293 i^{13}

Simplify

1 + 133 i + 234 i^{2} + 8583 i^{3} + 6435 i^{4} + 115833 i^{5} + 46332 i^{6} + 463323 i^{7} + 104247 i^{8} + 579153 i^{9} + 69498 i^{10} + 189543 i^{11} + 9477 i^{12} + 7293 i^{13} : 40963 i + 4096

= 40963 i + 4096

Rewrite in standard complex form:

4096 + 40963 i

= 4096 + 40963 i

(sin (x) + cos (x)) x

(1 + 3 i)^{11}

\frac{1}{2} (x - \frac{1}{10} sin (10 x))

π (\frac{π}{2} - 1)

(1 + e^{x})^{3}

What is (1+sqrt(3)i)^{13} ?
The solution to (1+sqrt(3)i)^{13} is 4096+4096sqrt(3)i