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Popular Algebra >

simplify (1+i)^{29}

Solution

simplify

(1 + i)^{29}

Solution

- 16384 - 16384 i

Solution steps

(1 + i)^{29}

Apply binomial theorem:

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

a = 1, b = i

= i = 0 \sum 29 (i 29) \cdot 1^{(29 - i)} i^{i}

Expand summation

= \frac{29 !}{0 ! ( 29 - 0 ) !} \cdot 1^{29} i^{0} + \frac{29 !}{1 ! ( 29 - 1 ) !} \cdot 1^{28} i^{1} + \frac{29 !}{2 ! ( 29 - 2 ) !} \cdot 1^{27} i^{2} + \frac{29 !}{3 ! ( 29 - 3 ) !} \cdot 1^{26} i^{3} + \frac{29 !}{4 ! ( 29 - 4 ) !} \cdot 1^{25} i^{4} + \frac{29 !}{5 ! ( 29 - 5 ) !} \cdot 1^{24} i^{5} + \frac{29 !}{6 ! ( 29 - 6 ) !} \cdot 1^{23} i^{6} + \frac{29 !}{7 ! ( 29 - 7 ) !} \cdot 1^{22} i^{7} + \frac{29 !}{8 ! ( 29 - 8 ) !} \cdot 1^{21} i^{8} + \frac{29 !}{9 ! ( 29 - 9 ) !} \cdot 1^{20} i^{9} + \frac{29 !}{10 ! ( 29 - 10 ) !} \cdot 1^{19} i^{10} + \frac{29 !}{11 ! ( 29 - 11 ) !} \cdot 1^{18} i^{11} + \frac{29 !}{12 ! ( 29 - 12 ) !} \cdot 1^{17} i^{12} + \frac{29 !}{13 ! ( 29 - 13 ) !} \cdot 1^{16} i^{13} + \frac{29 !}{14 ! ( 29 - 14 ) !} \cdot 1^{15} i^{14} + \frac{29 !}{15 ! ( 29 - 15 ) !} \cdot 1^{14} i^{15} + \frac{29 !}{16 ! ( 29 - 16 ) !} \cdot 1^{13} i^{16} + \frac{29 !}{17 ! ( 29 - 17 ) !} \cdot 1^{12} i^{17} + \frac{29 !}{18 ! ( 29 - 18 ) !} \cdot 1^{11} i^{18} + \frac{29 !}{19 ! ( 29 - 19 ) !} \cdot 1^{10} i^{19} + \frac{29 !}{20 ! ( 29 - 20 ) !} \cdot 1^{9} i^{20} + \frac{29 !}{21 ! ( 29 - 21 ) !} \cdot 1^{8} i^{21} + \frac{29 !}{22 ! ( 29 - 22 ) !} \cdot 1^{7} i^{22} + \frac{29 !}{23 ! ( 29 - 23 ) !} \cdot 1^{6} i^{23} + \frac{29 !}{24 ! ( 29 - 24 ) !} \cdot 1^{5} i^{24} + \frac{29 !}{25 ! ( 29 - 25 ) !} \cdot 1^{4} i^{25} + \frac{29 !}{26 ! ( 29 - 26 ) !} \cdot 1^{3} i^{26} + \frac{29 !}{27 ! ( 29 - 27 ) !} \cdot 1^{2} i^{27} + \frac{29 !}{28 ! ( 29 - 28 ) !} \cdot 1^{1} i^{28} + \frac{29 !}{29 ! ( 29 - 29 ) !} \cdot 1^{0} i^{29}

Simplify

\frac{29 !}{0 ! ( 29 - 0 ) !} \cdot 1^{29} i^{0} : 1

Simplify

\frac{29 !}{1 ! ( 29 - 1 ) !} \cdot 1^{28} i^{1} : 29 i

Simplify

\frac{29 !}{2 ! ( 29 - 2 ) !} \cdot 1^{27} i^{2} : 406 i^{2}

Simplify

\frac{29 !}{3 ! ( 29 - 3 ) !} \cdot 1^{26} i^{3} : 3654 i^{3}

Simplify

\frac{29 !}{4 ! ( 29 - 4 ) !} \cdot 1^{25} i^{4} : 23751 i^{4}

Simplify

\frac{29 !}{5 ! ( 29 - 5 ) !} \cdot 1^{24} i^{5} : 118755 i^{5}

Simplify

\frac{29 !}{6 ! ( 29 - 6 ) !} \cdot 1^{23} i^{6} : 475020 i^{6}

Simplify

\frac{29 !}{7 ! ( 29 - 7 ) !} \cdot 1^{22} i^{7} : 1560780 i^{7}

Simplify

\frac{29 !}{8 ! ( 29 - 8 ) !} \cdot 1^{21} i^{8} : 4292145 i^{8}

Simplify

\frac{29 !}{9 ! ( 29 - 9 ) !} \cdot 1^{20} i^{9} : 10015005 i^{9}

Simplify

\frac{29 !}{10 ! ( 29 - 10 ) !} \cdot 1^{19} i^{10} : 20030010 i^{10}

Simplify

\frac{29 !}{11 ! ( 29 - 11 ) !} \cdot 1^{18} i^{11} : 34597290 i^{11}

Simplify

\frac{29 !}{12 ! ( 29 - 12 ) !} \cdot 1^{17} i^{12} : 51895935 i^{12}

Simplify

\frac{29 !}{13 ! ( 29 - 13 ) !} \cdot 1^{16} i^{13} : 67863915 i^{13}

Simplify

\frac{29 !}{14 ! ( 29 - 14 ) !} \cdot 1^{15} i^{14} : 77558760 i^{14}

Simplify

\frac{29 !}{15 ! ( 29 - 15 ) !} \cdot 1^{14} i^{15} : 77558760 i^{15}

Simplify

\frac{29 !}{16 ! ( 29 - 16 ) !} \cdot 1^{13} i^{16} : 67863915 i^{16}

Simplify

\frac{29 !}{17 ! ( 29 - 17 ) !} \cdot 1^{12} i^{17} : 51895935 i^{17}

Simplify

\frac{29 !}{18 ! ( 29 - 18 ) !} \cdot 1^{11} i^{18} : 34597290 i^{18}

Simplify

\frac{29 !}{19 ! ( 29 - 19 ) !} \cdot 1^{10} i^{19} : 20030010 i^{19}

Simplify

\frac{29 !}{20 ! ( 29 - 20 ) !} \cdot 1^{9} i^{20} : 10015005 i^{20}

Simplify

\frac{29 !}{21 ! ( 29 - 21 ) !} \cdot 1^{8} i^{21} : 4292145 i^{21}

Simplify

\frac{29 !}{22 ! ( 29 - 22 ) !} \cdot 1^{7} i^{22} : 1560780 i^{22}

Simplify

\frac{29 !}{23 ! ( 29 - 23 ) !} \cdot 1^{6} i^{23} : 475020 i^{23}

Simplify

\frac{29 !}{24 ! ( 29 - 24 ) !} \cdot 1^{5} i^{24} : 118755 i^{24}

Simplify

\frac{29 !}{25 ! ( 29 - 25 ) !} \cdot 1^{4} i^{25} : 23751 i^{25}

Simplify

\frac{29 !}{26 ! ( 29 - 26 ) !} \cdot 1^{3} i^{26} : 3654 i^{26}

Simplify

\frac{29 !}{27 ! ( 29 - 27 ) !} \cdot 1^{2} i^{27} : 406 i^{27}

Simplify

\frac{29 !}{28 ! ( 29 - 28 ) !} \cdot 1^{1} i^{28} : 29 i^{28}

Simplify

\frac{29 !}{29 ! ( 29 - 29 ) !} \cdot 1^{0} i^{29} : i^{29}

= 1 + 29 i + 406 i^{2} + 3654 i^{3} + 23751 i^{4} + 118755 i^{5} + 475020 i^{6} + 1560780 i^{7} + 4292145 i^{8} + 10015005 i^{9} + 20030010 i^{10} + 34597290 i^{11} + 51895935 i^{12} + 67863915 i^{13} + 77558760 i^{14} + 77558760 i^{15} + 67863915 i^{16} + 51895935 i^{17} + 34597290 i^{18} + 20030010 i^{19} + 10015005 i^{20} + 4292145 i^{21} + 1560780 i^{22} + 475020 i^{23} + 118755 i^{24} + 23751 i^{25} + 3654 i^{26} + 406 i^{27} + 29 i^{28} + i^{29}

Simplify

1 + 29 i + 406 i^{2} + 3654 i^{3} + 23751 i^{4} + 118755 i^{5} + 475020 i^{6} + 1560780 i^{7} + 4292145 i^{8} + 10015005 i^{9} + 20030010 i^{10} + 34597290 i^{11} + 51895935 i^{12} + 67863915 i^{13} + 77558760 i^{14} + 77558760 i^{15} + 67863915 i^{16} + 51895935 i^{17} + 34597290 i^{18} + 20030010 i^{19} + 10015005 i^{20} + 4292145 i^{21} + 1560780 i^{22} + 475020 i^{23} + 118755 i^{24} + 23751 i^{25} + 3654 i^{26} + 406 i^{27} + 29 i^{28} + i^{29} : - 16384 i - 16384

= - 16384 i - 16384

Rewrite in standard complex form:

- 16384 - 16384 i

= - 16384 - 16384 i

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Frequently Asked Questions (FAQ)

What is (1+i)^{29} ?
The solution to (1+i)^{29} is -16384-16384i

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