What is (-1+sqrt(3)i)^{10} ?

The solution to (-1+sqrt(3)i)^{10} is -512+512sqrt(3)i

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simplify (-1+sqrt(3)i)^{10}

simplify

(- 1 + 3 i)^{10}

- 512 + 5123 i

Solution steps

(- 1 + 3 i)^{10}

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 10 (i 10) (- 1)^{(10 - i)} (3 i)^{i}

Expand summation

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

Simplify

\frac{10 !}{0 ! ( 10 - 0 ) !} (- 1)^{10} (3 i)^{0} : 1

Simplify

\frac{10 !}{1 ! ( 10 - 1 ) !} (- 1)^{9} (3 i)^{1} : - 103 i

Simplify

\frac{10 !}{2 ! ( 10 - 2 ) !} (- 1)^{8} (3 i)^{2} : 135 i^{2}

Simplify

\frac{10 !}{3 ! ( 10 - 3 ) !} (- 1)^{7} (3 i)^{3} : - 3603 i^{3}

Simplify

\frac{10 !}{4 ! ( 10 - 4 ) !} (- 1)^{6} (3 i)^{4} : 1890 i^{4}

Simplify

\frac{10 !}{5 ! ( 10 - 5 ) !} (- 1)^{5} (3 i)^{5} : - 22683 i^{5}

Simplify

\frac{10 !}{6 ! ( 10 - 6 ) !} (- 1)^{4} (3 i)^{6} : 5670 i^{6}

Simplify

\frac{10 !}{7 ! ( 10 - 7 ) !} (- 1)^{3} (3 i)^{7} : - 32403 i^{7}

Simplify

\frac{10 !}{8 ! ( 10 - 8 ) !} (- 1)^{2} (3 i)^{8} : 3645 i^{8}

Simplify

\frac{10 !}{9 ! ( 10 - 9 ) !} (- 1)^{1} (3 i)^{9} : - 8103 i^{9}

Simplify

\frac{10 !}{10 ! ( 10 - 10 ) !} (- 1)^{0} (3 i)^{10} : 243 i^{10}

= 1 - 103 i + 135 i^{2} - 3603 i^{3} + 1890 i^{4} - 22683 i^{5} + 5670 i^{6} - 32403 i^{7} + 3645 i^{8} - 8103 i^{9} + 243 i^{10}

Simplify

1 - 103 i + 135 i^{2} - 3603 i^{3} + 1890 i^{4} - 22683 i^{5} + 5670 i^{6} - 32403 i^{7} + 3645 i^{8} - 8103 i^{9} + 243 i^{10} : 5123 i - 512

= 5123 i - 512

Rewrite in standard complex form:

- 512 + 5123 i

= - 512 + 5123 i

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What is (-1+sqrt(3)i)^{10} ?
The solution to (-1+sqrt(3)i)^{10} is -512+512sqrt(3)i