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Popular Algebra >

expand (sqrt(3)-i)^{27}

Solution

expand

(3 - i)^{27}

Solution

\frac{2.28001 E 15 3 + 6.81911 E 15 i}{155925} - \frac{9.10546 E 19}{2075673600 3} - \frac{2.73164 E 20}{6227020800} i

Solution steps

(3 - i)^{27}

Apply binomial theorem:

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

a = 3, b = - i

= i = 0 \sum 27 (i 27) (3)^{(27 - i)} (- i)^{i}

Expand summation

= \frac{27 !}{0 ! ( 27 - 0 ) !} (3)^{27} (- i)^{0} + \frac{27 !}{1 ! ( 27 - 1 ) !} (3)^{26} (- i)^{1} + \frac{27 !}{2 ! ( 27 - 2 ) !} (3)^{25} (- i)^{2} + \frac{27 !}{3 ! ( 27 - 3 ) !} (3)^{24} (- i)^{3} + \frac{27 !}{4 ! ( 27 - 4 ) !} (3)^{23} (- i)^{4} + \frac{27 !}{5 ! ( 27 - 5 ) !} (3)^{22} (- i)^{5} + \frac{27 !}{6 ! ( 27 - 6 ) !} (3)^{21} (- i)^{6} + \frac{27 !}{7 ! ( 27 - 7 ) !} (3)^{20} (- i)^{7} + \frac{27 !}{8 ! ( 27 - 8 ) !} (3)^{19} (- i)^{8} + \frac{27 !}{9 ! ( 27 - 9 ) !} (3)^{18} (- i)^{9} + \frac{27 !}{10 ! ( 27 - 10 ) !} (3)^{17} (- i)^{10} + \frac{27 !}{11 ! ( 27 - 11 ) !} (3)^{16} (- i)^{11} + \frac{27 !}{12 ! ( 27 - 12 ) !} (3)^{15} (- i)^{12} + \frac{27 !}{13 ! ( 27 - 13 ) !} (3)^{14} (- i)^{13} + \frac{27 !}{14 ! ( 27 - 14 ) !} (3)^{13} (- i)^{14} + \frac{27 !}{15 ! ( 27 - 15 ) !} (3)^{12} (- i)^{15} + \frac{27 !}{16 ! ( 27 - 16 ) !} (3)^{11} (- i)^{16} + \frac{27 !}{17 ! ( 27 - 17 ) !} (3)^{10} (- i)^{17} + \frac{27 !}{18 ! ( 27 - 18 ) !} (3)^{9} (- i)^{18} + \frac{27 !}{19 ! ( 27 - 19 ) !} (3)^{8} (- i)^{19} + \frac{27 !}{20 ! ( 27 - 20 ) !} (3)^{7} (- i)^{20} + \frac{27 !}{21 ! ( 27 - 21 ) !} (3)^{6} (- i)^{21} + \frac{27 !}{22 ! ( 27 - 22 ) !} (3)^{5} (- i)^{22} + \frac{27 !}{23 ! ( 27 - 23 ) !} (3)^{4} (- i)^{23} + \frac{27 !}{24 ! ( 27 - 24 ) !} (3)^{3} (- i)^{24} + \frac{27 !}{25 ! ( 27 - 25 ) !} (3)^{2} (- i)^{25} + \frac{27 !}{26 ! ( 27 - 26 ) !} (3)^{1} (- i)^{26} + \frac{27 !}{27 ! ( 27 - 27 ) !} (3)^{0} (- i)^{27}

Simplify

\frac{27 !}{0 ! ( 27 - 0 ) !} (3)^{27} (- i)^{0} : 15943233

Simplify

\frac{27 !}{1 ! ( 27 - 1 ) !} (3)^{26} (- i)^{1} : - 43046721 i

Simplify

\frac{27 !}{2 ! ( 27 - 2 ) !} (3)^{25} (- i)^{2} : 1865357913 i^{2}

Simplify

\frac{27 !}{3 ! ( 27 - 3 ) !} (3)^{24} (- i)^{3} : - 1554464925 i^{3}

Simplify

\frac{27 !}{4 ! ( 27 - 4 ) !} (3)^{23} (- i)^{4} : 31089298503 i^{4}

Simplify

\frac{27 !}{5 ! ( 27 - 5 ) !} (3)^{22} (- i)^{5} : - 14301077310 i^{5}

Simplify

\frac{27 !}{6 ! ( 27 - 6 ) !} (3)^{21} (- i)^{6} : 174790944903 i^{6}

Simplify

\frac{27 !}{7 ! ( 27 - 7 ) !} (3)^{20} (- i)^{7} : - 52437283470 i^{7}

Simplify

\frac{27 !}{8 ! ( 27 - 8 ) !} (3)^{19} (- i)^{8} : 436977362253 i^{8}

Simplify

\frac{27 !}{9 ! ( 27 - 9 ) !} (3)^{18} (- i)^{9} : - 92250776475 i^{9}

Simplify

\frac{27 !}{10 ! ( 27 - 10 ) !} (3)^{17} (- i)^{10} : 553504658853 i^{10}

Simplify

\frac{27 !}{11 ! ( 27 - 11 ) !} (3)^{16} (- i)^{11} : - \frac{1.33381 E 16 i ^{11}}{155925}

Simplify

\frac{27 !}{12 ! ( 27 - 12 ) !} (3)^{15} (- i)^{12} : \frac{3 \cdot 1.82109 E 19 i ^{12}}{479001600}

Simplify

\frac{27 !}{13 ! ( 27 - 13 ) !} (3)^{14} (- i)^{13} : - \frac{2.73164 E 20 i ^{13}}{6227020800}

Simplify

\frac{27 !}{14 ! ( 27 - 14 ) !} (3)^{13} (- i)^{14} : \frac{3 \cdot 9.10546 E 19 i ^{14}}{6227020800}

Simplify

\frac{27 !}{15 ! ( 27 - 15 ) !} (3)^{12} (- i)^{15} : - 12672833940 i^{15}

Simplify

\frac{27 !}{16 ! ( 27 - 16 ) !} (3)^{11} (- i)^{16} : 31682084853 i^{16}

Simplify

\frac{27 !}{17 ! ( 27 - 17 ) !} (3)^{10} (- i)^{17} : - 2050017255 i^{17}

Simplify

\frac{27 !}{18 ! ( 27 - 18 ) !} (3)^{9} (- i)^{18} : 3796328253 i^{18}

Simplify

\frac{27 !}{19 ! ( 27 - 19 ) !} (3)^{8} (- i)^{19} : - 179826075 i^{19}

Simplify

\frac{27 !}{20 ! ( 27 - 20 ) !} (3)^{7} (- i)^{20} : 239768103 i^{20}

Simplify

\frac{27 !}{21 ! ( 27 - 21 ) !} (3)^{6} (- i)^{21} : - 7992270 i^{21}

Simplify

\frac{27 !}{22 ! ( 27 - 22 ) !} (3)^{5} (- i)^{22} : 7265703 i^{22}

Simplify

\frac{27 !}{23 ! ( 27 - 23 ) !} (3)^{4} (- i)^{23} : - 157950 i^{23}

Simplify

\frac{27 !}{24 ! ( 27 - 24 ) !} (3)^{3} (- i)^{24} : 87753 i^{24}

Simplify

\frac{27 !}{25 ! ( 27 - 25 ) !} (3)^{2} (- i)^{25} : - 1053 i^{25}

Simplify

\frac{27 !}{26 ! ( 27 - 26 ) !} (3)^{1} (- i)^{26} : 273 i^{26}

Simplify

\frac{27 !}{27 ! ( 27 - 27 ) !} (3)^{0} (- i)^{27} : - i^{27}

= 15943233 - 43046721 i + 1865357913 i^{2} - 1554464925 i^{3} + 31089298503 i^{4} - 14301077310 i^{5} + 174790944903 i^{6} - 52437283470 i^{7} + 436977362253 i^{8} - 92250776475 i^{9} + 553504658853 i^{10} - \frac{1.33381 E 16 i ^{11}}{155925} + \frac{3 \cdot 1.82109 E 19 i ^{12}}{479001600} - \frac{2.73164 E 20 i ^{13}}{6227020800} + \frac{3 \cdot 9.10546 E 19 i ^{14}}{6227020800} - 12672833940 i^{15} + 31682084853 i^{16} - 2050017255 i^{17} + 3796328253 i^{18} - 179826075 i^{19} + 239768103 i^{20} - 7992270 i^{21} + 7265703 i^{22} - 157950 i^{23} + 87753 i^{24} - 1053 i^{25} + 273 i^{26} - i^{27}

Simplify

15943233 - 43046721 i + 1865357913 i^{2} - 1554464925 i^{3} + 31089298503 i^{4} - 14301077310 i^{5} + 174790944903 i^{6} - 52437283470 i^{7} + 436977362253 i^{8} - 92250776475 i^{9} + 553504658853 i^{10} - \frac{1.33381 E 16 i ^{11}}{155925} + \frac{3 \cdot 1.82109 E 19 i ^{12}}{479001600} - \frac{2.73164 E 20 i ^{13}}{6227020800} + \frac{3 \cdot 9.10546 E 19 i ^{14}}{6227020800} - 12672833940 i^{15} + 31682084853 i^{16} - 2050017255 i^{17} + 3796328253 i^{18} - 179826075 i^{19} + 239768103 i^{20} - 7992270 i^{21} + 7265703 i^{22} - 157950 i^{23} + 87753 i^{24} - 1053 i^{25} + 273 i^{26} - i^{27} : \frac{6.81911 E 15 i + 2.28001 E 15 3}{155925} - \frac{2.73164 E 20 i}{6227020800} - \frac{9.10546 E 19}{2075673600 3}

= \frac{6.81911 E 15 i + 2.28001 E 15 3}{155925} - \frac{2.73164 E 20 i}{6227020800} - \frac{9.10546 E 19}{2075673600 3}

Rewrite in standard complex form:

\frac{2.28001 E 15 3 + 6.81911 E 15 i}{155925} - \frac{9.10546 E 19}{2075673600 3} - \frac{2.73164 E 20}{6227020800} i

= \frac{2.28001 E 15 3 + 6.81911 E 15 i}{155925} - \frac{9.10546 E 19}{2075673600 3} - \frac{2.73164 E 20}{6227020800} i

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