What is expand (1-(sqrt(3-i))/2)^{24} ?

The solution to expand (1-(sqrt(3-i))/2)^{24} is

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expand (1-(sqrt(3-i))/2)^{24}

Solution

expand

(1 - \frac{3 - i}{2})^{24}

Solution

- 12 3 - i - 253 (3 - i)^{\frac{3}{2}} - \frac{156009 ( 3 - i ) ^{\frac{11}{2}}}{128} - \frac{156009 ( 3 - i ) ^{\frac{13}{2}}}{512} - \frac{81719 ( 3 - i ) ^{\frac{15}{2}}}{2048} - \frac{5313 ( 3 - i ) ^{\frac{19}{2}}}{65536} - \frac{43263 ( 3 - i ) ^{\frac{17}{2}}}{16384} - \frac{253 ( 3 - i ) ^{\frac{21}{2}}}{262144} - \frac{3 ( 3 - i ) ^{\frac{23}{2}}}{1048576} - \frac{5313 ( 3 - i ) ^{\frac{5}{2}}}{4} - \frac{43263 ( 3 - i ) ^{\frac{7}{2}}}{16} - \frac{81719 ( 3 - i ) ^{\frac{9}{2}}}{32} - \frac{7847256981 i}{4096} - \frac{126253886913}{262144} + 208 - 69 i

Solution steps

(1 - \frac{3 - i}{2})^{24}

Apply binomial theorem:

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

a = 1, b = - \frac{3 - i}{2}

= i = 0 \sum 24 (i 24) \cdot 1^{(24 - i)} (- \frac{3 - i}{2})^{i}

Expand summation

= \frac{24 !}{0 ! ( 24 - 0 ) !} \cdot 1^{24} (- \frac{3 - i}{2})^{0} + \frac{24 !}{1 ! ( 24 - 1 ) !} \cdot 1^{23} (- \frac{3 - i}{2})^{1} + \frac{24 !}{2 ! ( 24 - 2 ) !} \cdot 1^{22} (- \frac{3 - i}{2})^{2} + \frac{24 !}{3 ! ( 24 - 3 ) !} \cdot 1^{21} (- \frac{3 - i}{2})^{3} + \frac{24 !}{4 ! ( 24 - 4 ) !} \cdot 1^{20} (- \frac{3 - i}{2})^{4} + \frac{24 !}{5 ! ( 24 - 5 ) !} \cdot 1^{19} (- \frac{3 - i}{2})^{5} + \frac{24 !}{6 ! ( 24 - 6 ) !} \cdot 1^{18} (- \frac{3 - i}{2})^{6} + \frac{24 !}{7 ! ( 24 - 7 ) !} \cdot 1^{17} (- \frac{3 - i}{2})^{7} + \frac{24 !}{8 ! ( 24 - 8 ) !} \cdot 1^{16} (- \frac{3 - i}{2})^{8} + \frac{24 !}{9 ! ( 24 - 9 ) !} \cdot 1^{15} (- \frac{3 - i}{2})^{9} + \frac{24 !}{10 ! ( 24 - 10 ) !} \cdot 1^{14} (- \frac{3 - i}{2})^{10} + \frac{24 !}{11 ! ( 24 - 11 ) !} \cdot 1^{13} (- \frac{3 - i}{2})^{11} + \frac{24 !}{12 ! ( 24 - 12 ) !} \cdot 1^{12} (- \frac{3 - i}{2})^{12} + \frac{24 !}{13 ! ( 24 - 13 ) !} \cdot 1^{11} (- \frac{3 - i}{2})^{13} + \frac{24 !}{14 ! ( 24 - 14 ) !} \cdot 1^{10} (- \frac{3 - i}{2})^{14} + \frac{24 !}{15 ! ( 24 - 15 ) !} \cdot 1^{9} (- \frac{3 - i}{2})^{15} + \frac{24 !}{16 ! ( 24 - 16 ) !} \cdot 1^{8} (- \frac{3 - i}{2})^{16} + \frac{24 !}{17 ! ( 24 - 17 ) !} \cdot 1^{7} (- \frac{3 - i}{2})^{17} + \frac{24 !}{18 ! ( 24 - 18 ) !} \cdot 1^{6} (- \frac{3 - i}{2})^{18} + \frac{24 !}{19 ! ( 24 - 19 ) !} \cdot 1^{5} (- \frac{3 - i}{2})^{19} + \frac{24 !}{20 ! ( 24 - 20 ) !} \cdot 1^{4} (- \frac{3 - i}{2})^{20} + \frac{24 !}{21 ! ( 24 - 21 ) !} \cdot 1^{3} (- \frac{3 - i}{2})^{21} + \frac{24 !}{22 ! ( 24 - 22 ) !} \cdot 1^{2} (- \frac{3 - i}{2})^{22} + \frac{24 !}{23 ! ( 24 - 23 ) !} \cdot 1^{1} (- \frac{3 - i}{2})^{23} + \frac{24 !}{24 ! ( 24 - 24 ) !} \cdot 1^{0} (- \frac{3 - i}{2})^{24}

\frac{24 !}{0 ! ( 24 - 0 ) !} \cdot 1^{24} (- \frac{3 - i}{2})^{0} = 1

Simplify

\frac{24 !}{1 ! ( 24 - 1 ) !} \cdot 1^{23} (- \frac{3 - i}{2})^{1} : - 12 3 - i

Simplify

\frac{24 !}{2 ! ( 24 - 2 ) !} \cdot 1^{22} (- \frac{3 - i}{2})^{2} : 69 (3 - i)

Simplify

\frac{24 !}{3 ! ( 24 - 3 ) !} \cdot 1^{21} (- \frac{3 - i}{2})^{3} : - 253 (3 - i)^{\frac{3}{2}}

Simplify

\frac{24 !}{4 ! ( 24 - 4 ) !} \cdot 1^{20} (- \frac{3 - i}{2})^{4} : \frac{5313 ( 4 - 3 i )}{4}

Simplify

\frac{24 !}{5 ! ( 24 - 5 ) !} \cdot 1^{19} (- \frac{3 - i}{2})^{5} : - \frac{5313 ( 3 - i ) ^{\frac{5}{2}}}{4}

Simplify

\frac{24 !}{6 ! ( 24 - 6 ) !} \cdot 1^{18} (- \frac{3 - i}{2})^{6} : \frac{33649 ( 9 - 13 i )}{8}

Simplify

\frac{24 !}{7 ! ( 24 - 7 ) !} \cdot 1^{17} (- \frac{3 - i}{2})^{7} : - \frac{43263 ( 3 - i ) ^{\frac{7}{2}}}{16}

Simplify

\frac{24 !}{8 ! ( 24 - 8 ) !} \cdot 1^{16} (- \frac{3 - i}{2})^{8} : \frac{735471 ( 3 - i ) ^{4}}{256}

Simplify

\frac{24 !}{9 ! ( 24 - 9 ) !} \cdot 1^{15} (- \frac{3 - i}{2})^{9} : - \frac{81719 ( 3 - i ) ^{\frac{9}{2}}}{32}

Simplify

\frac{24 !}{10 ! ( 24 - 10 ) !} \cdot 1^{14} (- \frac{3 - i}{2})^{10} : \frac{245157 ( 3 - i ) ^{5}}{128}

Simplify

\frac{24 !}{11 ! ( 24 - 11 ) !} \cdot 1^{13} (- \frac{3 - i}{2})^{11} : - \frac{156009 ( 3 - i ) ^{\frac{11}{2}}}{128}

Simplify

\frac{24 !}{12 ! ( 24 - 12 ) !} \cdot 1^{12} (- \frac{3 - i}{2})^{12} : \frac{676039 ( 3 - i ) ^{6}}{1024}

Simplify

\frac{24 !}{13 ! ( 24 - 13 ) !} \cdot 1^{11} (- \frac{3 - i}{2})^{13} : - \frac{156009 ( 3 - i ) ^{\frac{13}{2}}}{512}

Simplify

\frac{24 !}{14 ! ( 24 - 14 ) !} \cdot 1^{10} (- \frac{3 - i}{2})^{14} : \frac{245157 ( 3 - i ) ^{7}}{2048}

Simplify

\frac{24 !}{15 ! ( 24 - 15 ) !} \cdot 1^{9} (- \frac{3 - i}{2})^{15} : - \frac{81719 ( 3 - i ) ^{\frac{15}{2}}}{2048}

Simplify

\frac{24 !}{16 ! ( 24 - 16 ) !} \cdot 1^{8} (- \frac{3 - i}{2})^{16} : \frac{735471 ( 3 - i ) ^{8}}{65536}

Simplify

\frac{24 !}{17 ! ( 24 - 17 ) !} \cdot 1^{7} (- \frac{3 - i}{2})^{17} : - \frac{43263 ( 3 - i ) ^{\frac{17}{2}}}{16384}

Simplify

\frac{24 !}{18 ! ( 24 - 18 ) !} \cdot 1^{6} (- \frac{3 - i}{2})^{18} : \frac{33649 ( 3 - i ) ^{9}}{65536}

Simplify

\frac{24 !}{19 ! ( 24 - 19 ) !} \cdot 1^{5} (- \frac{3 - i}{2})^{19} : - \frac{5313 ( 3 - i ) ^{\frac{19}{2}}}{65536}

Simplify

\frac{24 !}{20 ! ( 24 - 20 ) !} \cdot 1^{4} (- \frac{3 - i}{2})^{20} : \frac{5313 ( 3 - i ) ^{10}}{524288}

Simplify

\frac{24 !}{21 ! ( 24 - 21 ) !} \cdot 1^{3} (- \frac{3 - i}{2})^{21} : - \frac{253 ( 3 - i ) ^{\frac{21}{2}}}{262144}

Simplify

\frac{24 !}{22 ! ( 24 - 22 ) !} \cdot 1^{2} (- \frac{3 - i}{2})^{22} : \frac{69 ( 3 - i ) ^{11}}{1048576}

Simplify

\frac{24 !}{23 ! ( 24 - 23 ) !} \cdot 1^{1} (- \frac{3 - i}{2})^{23} : - \frac{3 ( 3 - i ) ^{\frac{23}{2}}}{1048576}

Simplify

\frac{24 !}{24 ! ( 24 - 24 ) !} \cdot 1^{0} (- \frac{3 - i}{2})^{24} : \frac{( 3 - i ) ^{12}}{16777216}

= 1 - 12 3 - i + 69 (3 - i) - 253 (3 - i)^{\frac{3}{2}} + \frac{5313 ( 4 - 3 i )}{4} - \frac{5313 ( 3 - i ) ^{\frac{5}{2}}}{4} + \frac{33649 ( 9 - 13 i )}{8} - \frac{43263 ( 3 - i ) ^{\frac{7}{2}}}{16} + \frac{735471 ( 3 - i ) ^{4}}{256} - \frac{81719 ( 3 - i ) ^{\frac{9}{2}}}{32} + \frac{245157 ( 3 - i ) ^{5}}{128} - \frac{156009 ( 3 - i ) ^{\frac{11}{2}}}{128} + \frac{676039 ( 3 - i ) ^{6}}{1024} - \frac{156009 ( 3 - i ) ^{\frac{13}{2}}}{512} + \frac{245157 ( 3 - i ) ^{7}}{2048} - \frac{81719 ( 3 - i ) ^{\frac{15}{2}}}{2048} + \frac{735471 ( 3 - i ) ^{8}}{65536} - \frac{43263 ( 3 - i ) ^{\frac{17}{2}}}{16384} + \frac{33649 ( 3 - i ) ^{9}}{65536} - \frac{5313 ( 3 - i ) ^{\frac{19}{2}}}{65536} + \frac{5313 ( 3 - i ) ^{10}}{524288} - \frac{253 ( 3 - i ) ^{\frac{21}{2}}}{262144} + \frac{69 ( 3 - i ) ^{11}}{1048576} - \frac{3 ( 3 - i ) ^{\frac{23}{2}}}{1048576} + \frac{( 3 - i ) ^{12}}{16777216}

Simplify

1 - 12 3 - i + 69 (3 - i) - 253 (3 - i)^{\frac{3}{2}} + \frac{5313 ( 4 - 3 i )}{4} - \frac{5313 ( 3 - i ) ^{\frac{5}{2}}}{4} + \frac{33649 ( 9 - 13 i )}{8} - \frac{43263 ( 3 - i ) ^{\frac{7}{2}}}{16} + \frac{735471 ( 3 - i ) ^{4}}{256} - \frac{81719 ( 3 - i ) ^{\frac{9}{2}}}{32} + \frac{245157 ( 3 - i ) ^{5}}{128} - \frac{156009 ( 3 - i ) ^{\frac{11}{2}}}{128} + \frac{676039 ( 3 - i ) ^{6}}{1024} - \frac{156009 ( 3 - i ) ^{\frac{13}{2}}}{512} + \frac{245157 ( 3 - i ) ^{7}}{2048} - \frac{81719 ( 3 - i ) ^{\frac{15}{2}}}{2048} + \frac{735471 ( 3 - i ) ^{8}}{65536} - \frac{43263 ( 3 - i ) ^{\frac{17}{2}}}{16384} + \frac{33649 ( 3 - i ) ^{9}}{65536} - \frac{5313 ( 3 - i ) ^{\frac{19}{2}}}{65536} + \frac{5313 ( 3 - i ) ^{10}}{524288} - \frac{253 ( 3 - i ) ^{\frac{21}{2}}}{262144} + \frac{69 ( 3 - i ) ^{11}}{1048576} - \frac{3 ( 3 - i ) ^{\frac{23}{2}}}{1048576} + \frac{( 3 - i ) ^{12}}{16777216} : - 12 3 - i - 253 (3 - i)^{\frac{3}{2}} + \frac{- 1278025728 ( 3 - i ) ^{\frac{11}{2}} - 319506432 ( 3 - i ) ^{\frac{13}{2}} - 41840128 ( 3 - i ) ^{\frac{15}{2}} - 85008 ( 3 - i ) ^{\frac{19}{2}} - 2768832 ( 3 - i ) ^{\frac{17}{2}} - 1012 ( 3 - i ) ^{\frac{21}{2}} - 3 ( 3 - i ) ^{\frac{23}{2}} - 1392771072 ( 3 - i ) ^{\frac{5}{2}} - 2835283968 ( 3 - i ) ^{\frac{7}{2}} - 2677768192 ( 3 - i ) ^{\frac{9}{2}} - 2008897787136 i - 505015547652}{1048576} + 208 - 69 i

= - 12 3 - i - 253 (3 - i)^{\frac{3}{2}} + \frac{- 1278025728 ( 3 - i ) ^{\frac{11}{2}} - 319506432 ( 3 - i ) ^{\frac{13}{2}} - 41840128 ( 3 - i ) ^{\frac{15}{2}} - 85008 ( 3 - i ) ^{\frac{19}{2}} - 2768832 ( 3 - i ) ^{\frac{17}{2}} - 1012 ( 3 - i ) ^{\frac{21}{2}} - 3 ( 3 - i ) ^{\frac{23}{2}} - 1392771072 ( 3 - i ) ^{\frac{5}{2}} - 2835283968 ( 3 - i ) ^{\frac{7}{2}} - 2677768192 ( 3 - i ) ^{\frac{9}{2}} - 2008897787136 i - 505015547652}{1048576} + 208 - 69 i

Apply the fraction rule:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{- 1278025728 ( 3 - i ) ^{\frac{11}{2}} - 319506432 ( 3 - i ) ^{\frac{13}{2}} - 41840128 ( 3 - i ) ^{\frac{15}{2}} - 85008 ( 3 - i ) ^{\frac{19}{2}} - 2768832 ( 3 - i ) ^{\frac{17}{2}} - 1012 ( 3 - i ) ^{\frac{21}{2}} - 3 ( 3 - i ) ^{\frac{23}{2}} - 1392771072 ( 3 - i ) ^{\frac{5}{2}} - 2835283968 ( 3 - i ) ^{\frac{7}{2}} - 2677768192 ( 3 - i ) ^{\frac{9}{2}} - 2008897787136 i - 505015547652}{1048576} = - \frac{1278025728 ( 3 - i ) ^{\frac{11}{2}}}{1048576} - \frac{319506432 ( 3 - i ) ^{\frac{13}{2}}}{1048576} - \frac{41840128 ( 3 - i ) ^{\frac{15}{2}}}{1048576} - \frac{85008 ( 3 - i ) ^{\frac{19}{2}}}{1048576} - \frac{2768832 ( 3 - i ) ^{\frac{17}{2}}}{1048576} - \frac{1012 ( 3 - i ) ^{\frac{21}{2}}}{1048576} - \frac{3 ( 3 - i ) ^{\frac{23}{2}}}{1048576} - \frac{1392771072 ( 3 - i ) ^{\frac{5}{2}}}{1048576} - \frac{2835283968 ( 3 - i ) ^{\frac{7}{2}}}{1048576} - \frac{2677768192 ( 3 - i ) ^{\frac{9}{2}}}{1048576} - \frac{2008897787136 i}{1048576} - \frac{505015547652}{1048576}

= - 12 3 - i - 253 (3 - i)^{\frac{3}{2}} - \frac{1278025728 ( 3 - i ) ^{\frac{11}{2}}}{1048576} - \frac{319506432 ( 3 - i ) ^{\frac{13}{2}}}{1048576} - \frac{41840128 ( 3 - i ) ^{\frac{15}{2}}}{1048576} - \frac{85008 ( 3 - i ) ^{\frac{19}{2}}}{1048576} - \frac{2768832 ( 3 - i ) ^{\frac{17}{2}}}{1048576} - \frac{1012 ( 3 - i ) ^{\frac{21}{2}}}{1048576} - \frac{3 ( 3 - i ) ^{\frac{23}{2}}}{1048576} - \frac{1392771072 ( 3 - i ) ^{\frac{5}{2}}}{1048576} - \frac{2835283968 ( 3 - i ) ^{\frac{7}{2}}}{1048576} - \frac{2677768192 ( 3 - i ) ^{\frac{9}{2}}}{1048576} - \frac{2008897787136 i}{1048576} - \frac{505015547652}{1048576} + 208 - 69 i

\frac{1278025728 ( 3 - i ) ^{\frac{11}{2}}}{1048576} = \frac{156009 ( 3 - i ) ^{\frac{11}{2}}}{128}

\frac{319506432 ( 3 - i ) ^{\frac{13}{2}}}{1048576} = \frac{156009 ( 3 - i ) ^{\frac{13}{2}}}{512}

\frac{41840128 ( 3 - i ) ^{\frac{15}{2}}}{1048576} = \frac{81719 ( 3 - i ) ^{\frac{15}{2}}}{2048}

\frac{85008 ( 3 - i ) ^{\frac{19}{2}}}{1048576} = \frac{5313 ( 3 - i ) ^{\frac{19}{2}}}{65536}

\frac{2768832 ( 3 - i ) ^{\frac{17}{2}}}{1048576} = \frac{43263 ( 3 - i ) ^{\frac{17}{2}}}{16384}

\frac{1012 ( 3 - i ) ^{\frac{21}{2}}}{1048576} = \frac{253 ( 3 - i ) ^{\frac{21}{2}}}{262144}

\frac{1392771072 ( 3 - i ) ^{\frac{5}{2}}}{1048576} = \frac{5313 ( 3 - i ) ^{\frac{5}{2}}}{4}

\frac{2835283968 ( 3 - i ) ^{\frac{7}{2}}}{1048576} = \frac{43263 ( 3 - i ) ^{\frac{7}{2}}}{16}

\frac{2677768192 ( 3 - i ) ^{\frac{9}{2}}}{1048576} = \frac{81719 ( 3 - i ) ^{\frac{9}{2}}}{32}

\frac{2008897787136 i}{1048576} = \frac{7847256981 i}{4096}

\frac{505015547652}{1048576} = \frac{126253886913}{262144}

= - 12 3 - i - 253 (3 - i)^{\frac{3}{2}} - \frac{156009 ( 3 - i ) ^{\frac{11}{2}}}{128} - \frac{156009 ( 3 - i ) ^{\frac{13}{2}}}{512} - \frac{81719 ( 3 - i ) ^{\frac{15}{2}}}{2048} - \frac{5313 ( 3 - i ) ^{\frac{19}{2}}}{65536} - \frac{43263 ( 3 - i ) ^{\frac{17}{2}}}{16384} - \frac{253 ( 3 - i ) ^{\frac{21}{2}}}{262144} - \frac{3 ( 3 - i ) ^{\frac{23}{2}}}{1048576} - \frac{5313 ( 3 - i ) ^{\frac{5}{2}}}{4} - \frac{43263 ( 3 - i ) ^{\frac{7}{2}}}{16} - \frac{81719 ( 3 - i ) ^{\frac{9}{2}}}{32} - \frac{7847256981 i}{4096} - \frac{126253886913}{262144} + 208 - 69 i

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Frequently Asked Questions (FAQ)

What is expand (1-(sqrt(3-i))/2)^{24} ?
The solution to expand (1-(sqrt(3-i))/2)^{24} is