What is (-sqrt(3)+i)^{20} ?

The solution to (-sqrt(3)+i)^{20} is -524288+524288sqrt(3)i

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Popular >

simplify (-sqrt(3)+i)^{20}

Solution

simplify

(- 3 + i)^{20}

Solution

- 524288 + 5242883 i

Solution steps

(- 3 + i)^{20}

Apply binomial theorem:

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

a = - 3, b = i

= i = 0 \sum 20 (i 20) (- 3)^{(20 - i)} i^{i}

Expand summation

= \frac{20 !}{0 ! ( 20 - 0 ) !} (- 3)^{20} i^{0} + \frac{20 !}{1 ! ( 20 - 1 ) !} (- 3)^{19} i^{1} + \frac{20 !}{2 ! ( 20 - 2 ) !} (- 3)^{18} i^{2} + \frac{20 !}{3 ! ( 20 - 3 ) !} (- 3)^{17} i^{3} + \frac{20 !}{4 ! ( 20 - 4 ) !} (- 3)^{16} i^{4} + \frac{20 !}{5 ! ( 20 - 5 ) !} (- 3)^{15} i^{5} + \frac{20 !}{6 ! ( 20 - 6 ) !} (- 3)^{14} i^{6} + \frac{20 !}{7 ! ( 20 - 7 ) !} (- 3)^{13} i^{7} + \frac{20 !}{8 ! ( 20 - 8 ) !} (- 3)^{12} i^{8} + \frac{20 !}{9 ! ( 20 - 9 ) !} (- 3)^{11} i^{9} + \frac{20 !}{10 ! ( 20 - 10 ) !} (- 3)^{10} i^{10} + \frac{20 !}{11 ! ( 20 - 11 ) !} (- 3)^{9} i^{11} + \frac{20 !}{12 ! ( 20 - 12 ) !} (- 3)^{8} i^{12} + \frac{20 !}{13 ! ( 20 - 13 ) !} (- 3)^{7} i^{13} + \frac{20 !}{14 ! ( 20 - 14 ) !} (- 3)^{6} i^{14} + \frac{20 !}{15 ! ( 20 - 15 ) !} (- 3)^{5} i^{15} + \frac{20 !}{16 ! ( 20 - 16 ) !} (- 3)^{4} i^{16} + \frac{20 !}{17 ! ( 20 - 17 ) !} (- 3)^{3} i^{17} + \frac{20 !}{18 ! ( 20 - 18 ) !} (- 3)^{2} i^{18} + \frac{20 !}{19 ! ( 20 - 19 ) !} (- 3)^{1} i^{19} + \frac{20 !}{20 ! ( 20 - 20 ) !} (- 3)^{0} i^{20}

Simplify

\frac{20 !}{0 ! ( 20 - 0 ) !} (- 3)^{20} i^{0} : 59049

Simplify

\frac{20 !}{1 ! ( 20 - 1 ) !} (- 3)^{19} i^{1} : - 3936603 i

Simplify

\frac{20 !}{2 ! ( 20 - 2 ) !} (- 3)^{18} i^{2} : 3739770 i^{2}

Simplify

\frac{20 !}{3 ! ( 20 - 3 ) !} (- 3)^{17} i^{3} : - 74795403 i^{3}

Simplify

\frac{20 !}{4 ! ( 20 - 4 ) !} (- 3)^{16} i^{4} : 31788045 i^{4}

Simplify

\frac{20 !}{5 ! ( 20 - 5 ) !} (- 3)^{15} i^{5} : - 339072483 i^{5}

Simplify

\frac{20 !}{6 ! ( 20 - 6 ) !} (- 3)^{14} i^{6} : 84768120 i^{6}

Simplify

\frac{20 !}{7 ! ( 20 - 7 ) !} (- 3)^{13} i^{7} : - 565120803 i^{7}

Simplify

\frac{20 !}{8 ! ( 20 - 8 ) !} (- 3)^{12} i^{8} : 91832130 i^{8}

Simplify

\frac{20 !}{9 ! ( 20 - 9 ) !} (- 3)^{11} i^{9} : - 408142803 i^{9}

Simplify

\frac{20 !}{10 ! ( 20 - 10 ) !} (- 3)^{10} i^{10} : 44895708 i^{10}

Simplify

\frac{20 !}{11 ! ( 20 - 11 ) !} (- 3)^{9} i^{11} : - 136047603 i^{11}

Simplify

\frac{20 !}{12 ! ( 20 - 12 ) !} (- 3)^{8} i^{12} : 10203570 i^{12}

Simplify

\frac{20 !}{13 ! ( 20 - 13 ) !} (- 3)^{7} i^{13} : - 20930403 i^{13}

Simplify

\frac{20 !}{14 ! ( 20 - 14 ) !} (- 3)^{6} i^{14} : 1046520 i^{14}

Simplify

\frac{20 !}{15 ! ( 20 - 15 ) !} (- 3)^{5} i^{15} : - 1395363 i^{15}

Simplify

\frac{20 !}{16 ! ( 20 - 16 ) !} (- 3)^{4} i^{16} : 43605 i^{16}

Simplify

\frac{20 !}{17 ! ( 20 - 17 ) !} (- 3)^{3} i^{17} : - 34203 i^{17}

Simplify

\frac{20 !}{18 ! ( 20 - 18 ) !} (- 3)^{2} i^{18} : 570 i^{18}

Simplify

\frac{20 !}{19 ! ( 20 - 19 ) !} (- 3)^{1} i^{19} : - 203 i^{19}

Simplify

\frac{20 !}{20 ! ( 20 - 20 ) !} (- 3)^{0} i^{20} : i^{20}

= 59049 - 3936603 i + 3739770 i^{2} - 74795403 i^{3} + 31788045 i^{4} - 339072483 i^{5} + 84768120 i^{6} - 565120803 i^{7} + 91832130 i^{8} - 408142803 i^{9} + 44895708 i^{10} - 136047603 i^{11} + 10203570 i^{12} - 20930403 i^{13} + 1046520 i^{14} - 1395363 i^{15} + 43605 i^{16} - 34203 i^{17} + 570 i^{18} - 203 i^{19} + i^{20}

Simplify

59049 - 3936603 i + 3739770 i^{2} - 74795403 i^{3} + 31788045 i^{4} - 339072483 i^{5} + 84768120 i^{6} - 565120803 i^{7} + 91832130 i^{8} - 408142803 i^{9} + 44895708 i^{10} - 136047603 i^{11} + 10203570 i^{12} - 20930403 i^{13} + 1046520 i^{14} - 1395363 i^{15} + 43605 i^{16} - 34203 i^{17} + 570 i^{18} - 203 i^{19} + i^{20} : 5242883 i - 524288

= 5242883 i - 524288

Rewrite in standard complex form:

- 524288 + 5242883 i

= - 524288 + 5242883 i

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Frequently Asked Questions (FAQ)

What is (-sqrt(3)+i)^{20} ?
The solution to (-sqrt(3)+i)^{20} is -524288+524288sqrt(3)i