What is (1+sqrt(3)i)^{11} ?

The solution to (1+sqrt(3)i)^{11} is 1024-1024sqrt(3)i

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simplify (1+sqrt(3)i)^{11}

simplify

(1 + 3 i)^{11}

1024 - 10243 i

Solution steps

(1 + 3 i)^{11}

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 11 (i 11) \cdot 1^{(11 - i)} (3 i)^{i}

Expand summation

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

Simplify

\frac{11 !}{0 ! ( 11 - 0 ) !} \cdot 1^{11} (3 i)^{0} : 1

Simplify

\frac{11 !}{1 ! ( 11 - 1 ) !} \cdot 1^{10} (3 i)^{1} : 113 i

Simplify

\frac{11 !}{2 ! ( 11 - 2 ) !} \cdot 1^{9} (3 i)^{2} : 165 i^{2}

Simplify

\frac{11 !}{3 ! ( 11 - 3 ) !} \cdot 1^{8} (3 i)^{3} : 4953 i^{3}

Simplify

\frac{11 !}{4 ! ( 11 - 4 ) !} \cdot 1^{7} (3 i)^{4} : 2970 i^{4}

Simplify

\frac{11 !}{5 ! ( 11 - 5 ) !} \cdot 1^{6} (3 i)^{5} : 41583 i^{5}

Simplify

\frac{11 !}{6 ! ( 11 - 6 ) !} \cdot 1^{5} (3 i)^{6} : 12474 i^{6}

Simplify

\frac{11 !}{7 ! ( 11 - 7 ) !} \cdot 1^{4} (3 i)^{7} : 89103 i^{7}

Simplify

\frac{11 !}{8 ! ( 11 - 8 ) !} \cdot 1^{3} (3 i)^{8} : 13365 i^{8}

Simplify

\frac{11 !}{9 ! ( 11 - 9 ) !} \cdot 1^{2} (3 i)^{9} : 44553 i^{9}

Simplify

\frac{11 !}{10 ! ( 11 - 10 ) !} \cdot 1^{1} (3 i)^{10} : 2673 i^{10}

Simplify

\frac{11 !}{11 ! ( 11 - 11 ) !} \cdot 1^{0} (3 i)^{11} : 2433 i^{11}

= 1 + 113 i + 165 i^{2} + 4953 i^{3} + 2970 i^{4} + 41583 i^{5} + 12474 i^{6} + 89103 i^{7} + 13365 i^{8} + 44553 i^{9} + 2673 i^{10} + 2433 i^{11}

Simplify

1 + 113 i + 165 i^{2} + 4953 i^{3} + 2970 i^{4} + 41583 i^{5} + 12474 i^{6} + 89103 i^{7} + 13365 i^{8} + 44553 i^{9} + 2673 i^{10} + 2433 i^{11} : - 10243 i + 1024

= - 10243 i + 1024

Rewrite in standard complex form:

1024 - 10243 i

= 1024 - 10243 i

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

π (\frac{π}{2} - 1)

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

(6 x y - 2 y^{2} + 1) d x

\frac{d}{d y} ((x) (cos (x) y + 3 x y^{2}))

What is (1+sqrt(3)i)^{11} ?
The solution to (1+sqrt(3)i)^{11} is 1024-1024sqrt(3)i