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

Expand summation

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

\frac{10 !}{10 ! ( 10 - 10 ) !} \cdot 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

(2 x - 1) d x + (3 y + 7) d y

\frac{( x - 1 ) ^{2}}{x ^{2} + 1}

(\frac{1}{2} + \frac{1}{2} cos (2 x))^{4}

(x - x y) d y

e^{x} (x + 2)

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