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Popular >

simplify (1+isqrt(3))^{27}

Solution

simplify

(1 + i 3)^{27}

Solution

- \frac{8.67805 E 19}{2075673600} - \frac{2.73164 E 20}{6227020800} + \frac{1.33381 E 16}{155925}

+1

Decimal Notation

- 134217728

Solution steps

(1 + i 3)^{27}

Apply binomial theorem:

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

a = 1, b = i 3

= i = 0 \sum 27 (i 27) \cdot 1^{(27 - i)} (i 3)^{i}

Expand summation

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

Simplify

\frac{27 !}{0 ! ( 27 - 0 ) !} \cdot 1^{27} (i 3)^{0} : 1

Simplify

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

Simplify

\frac{27 !}{2 ! ( 27 - 2 ) !} \cdot 1^{25} (i 3)^{2} : 1053 i^{2}

Simplify

\frac{27 !}{3 ! ( 27 - 3 ) !} \cdot 1^{24} (i 3)^{3} : 87753 i^{3}

Simplify

\frac{27 !}{4 ! ( 27 - 4 ) !} \cdot 1^{23} (i 3)^{4} : 157950 i^{4}

Simplify

\frac{27 !}{5 ! ( 27 - 5 ) !} \cdot 1^{22} (i 3)^{5} : 7265703 i^{5}

Simplify

\frac{27 !}{6 ! ( 27 - 6 ) !} \cdot 1^{21} (i 3)^{6} : 7992270 i^{6}

Simplify

\frac{27 !}{7 ! ( 27 - 7 ) !} \cdot 1^{20} (i 3)^{7} : 239768103 i^{7}

Simplify

\frac{27 !}{8 ! ( 27 - 8 ) !} \cdot 1^{19} (i 3)^{8} : 179826075 i^{8}

Simplify

\frac{27 !}{9 ! ( 27 - 9 ) !} \cdot 1^{18} (i 3)^{9} : 3796328253 i^{9}

Simplify

\frac{27 !}{10 ! ( 27 - 10 ) !} \cdot 1^{17} (i 3)^{10} : 2050017255 i^{10}

Simplify

\frac{27 !}{11 ! ( 27 - 11 ) !} \cdot 1^{16} (i 3)^{11} : 31682084853 i^{11}

Simplify

\frac{27 !}{12 ! ( 27 - 12 ) !} \cdot 1^{15} (i 3)^{12} : 12672833940 i^{12}

Simplify

\frac{27 !}{13 ! ( 27 - 13 ) !} \cdot 1^{14} (i 3)^{13} : \frac{3 \cdot 9.10546 E 19 i ^{13}}{6227020800}

Simplify

\frac{27 !}{14 ! ( 27 - 14 ) !} \cdot 1^{13} (i 3)^{14} : \frac{2.73164 E 20 i ^{14}}{6227020800}

Simplify

\frac{27 !}{15 ! ( 27 - 15 ) !} \cdot 1^{12} (i 3)^{15} : \frac{3 \cdot 1.82109 E 19 i ^{15}}{479001600}

Simplify

\frac{27 !}{16 ! ( 27 - 16 ) !} \cdot 1^{11} (i 3)^{16} : \frac{1.33381 E 16 i ^{16}}{155925}

Simplify

\frac{27 !}{17 ! ( 27 - 17 ) !} \cdot 1^{10} (i 3)^{17} : 553504658853 i^{17}

Simplify

\frac{27 !}{18 ! ( 27 - 18 ) !} \cdot 1^{9} (i 3)^{18} : 92250776475 i^{18}

Simplify

\frac{27 !}{19 ! ( 27 - 19 ) !} \cdot 1^{8} (i 3)^{19} : 436977362253 i^{19}

Simplify

\frac{27 !}{20 ! ( 27 - 20 ) !} \cdot 1^{7} (i 3)^{20} : 52437283470 i^{20}

Simplify

\frac{27 !}{21 ! ( 27 - 21 ) !} \cdot 1^{6} (i 3)^{21} : 174790944903 i^{21}

Simplify

\frac{27 !}{22 ! ( 27 - 22 ) !} \cdot 1^{5} (i 3)^{22} : 14301077310 i^{22}

Simplify

\frac{27 !}{23 ! ( 27 - 23 ) !} \cdot 1^{4} (i 3)^{23} : 31089298503 i^{23}

Simplify

\frac{27 !}{24 ! ( 27 - 24 ) !} \cdot 1^{3} (i 3)^{24} : 1554464925 i^{24}

Simplify

\frac{27 !}{25 ! ( 27 - 25 ) !} \cdot 1^{2} (i 3)^{25} : 1865357913 i^{25}

Simplify

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

Simplify

\frac{27 !}{27 ! ( 27 - 27 ) !} \cdot 1^{0} (i 3)^{27} : 15943233 i^{27}

= 1 + 273 i + 1053 i^{2} + 87753 i^{3} + 157950 i^{4} + 7265703 i^{5} + 7992270 i^{6} + 239768103 i^{7} + 179826075 i^{8} + 3796328253 i^{9} + 2050017255 i^{10} + 31682084853 i^{11} + 12672833940 i^{12} + \frac{3 \cdot 9.10546 E 19 i ^{13}}{6227020800} + \frac{2.73164 E 20 i ^{14}}{6227020800} + \frac{3 \cdot 1.82109 E 19 i ^{15}}{479001600} + \frac{1.33381 E 16 i ^{16}}{155925} + 553504658853 i^{17} + 92250776475 i^{18} + 436977362253 i^{19} + 52437283470 i^{20} + 174790944903 i^{21} + 14301077310 i^{22} + 31089298503 i^{23} + 1554464925 i^{24} + 1865357913 i^{25} + 43046721 i^{26} + 15943233 i^{27}

Simplify

1 + 273 i + 1053 i^{2} + 87753 i^{3} + 157950 i^{4} + 7265703 i^{5} + 7992270 i^{6} + 239768103 i^{7} + 179826075 i^{8} + 3796328253 i^{9} + 2050017255 i^{10} + 31682084853 i^{11} + 12672833940 i^{12} + \frac{3 \cdot 9.10546 E 19 i ^{13}}{6227020800} + \frac{2.73164 E 20 i ^{14}}{6227020800} + \frac{3 \cdot 1.82109 E 19 i ^{15}}{479001600} + \frac{1.33381 E 16 i ^{16}}{155925} + 553504658853 i^{17} + 92250776475 i^{18} + 436977362253 i^{19} + 52437283470 i^{20} + 174790944903 i^{21} + 14301077310 i^{22} + 31089298503 i^{23} + 1554464925 i^{24} + 1865357913 i^{25} + 43046721 i^{26} + 15943233 i^{27} : - \frac{8.67805 E 19}{2075673600} - \frac{2.73164 E 20}{6227020800} + \frac{1.33381 E 16}{155925}

= - \frac{8.67805 E 19}{2075673600} - \frac{2.73164 E 20}{6227020800} + \frac{1.33381 E 16}{155925}

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