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