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

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

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

simplify

(- 3 + i)^{10}

512 + 5123 i

Solution steps

(- 3 + i)^{10}

Apply binomial theorem:

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

a = - 3, b = i

= i = 0 \sum 10 (i 10) (- 3)^{(10 - i)} i^{i}

Expand summation

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

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

Simplify

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

= 5123 i + 512

Rewrite in standard complex form:

512 + 5123 i

= 512 + 5123 i

3 (2 - 510)

\frac{( x - 4 ) ( x - 1 ) ( x + 3 )}{( x - 3 ) ( x + 2 )}

(e - 1)^{2}

\frac{( 2 - x ) ^{3}}{3 ( x - 1 ) ^{2}}

\frac{1}{x ( x + 3 )}

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