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Popular Algebra >

simplify (1+i)^{22}

Solution

simplify

(1 + i)^{22}

Solution

- 2048 i

Solution steps

(1 + i)^{22}

Apply binomial theorem:

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

a = 1, b = i

= i = 0 \sum 22 (i 22) \cdot 1^{(22 - i)} i^{i}

Expand summation

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

Simplify

\frac{22 !}{0 ! ( 22 - 0 ) !} \cdot 1^{22} i^{0} : 1

Simplify

\frac{22 !}{1 ! ( 22 - 1 ) !} \cdot 1^{21} i^{1} : 22 i

Simplify

\frac{22 !}{2 ! ( 22 - 2 ) !} \cdot 1^{20} i^{2} : 231 i^{2}

Simplify

\frac{22 !}{3 ! ( 22 - 3 ) !} \cdot 1^{19} i^{3} : 1540 i^{3}

Simplify

\frac{22 !}{4 ! ( 22 - 4 ) !} \cdot 1^{18} i^{4} : 7315 i^{4}

Simplify

\frac{22 !}{5 ! ( 22 - 5 ) !} \cdot 1^{17} i^{5} : 26334 i^{5}

Simplify

\frac{22 !}{6 ! ( 22 - 6 ) !} \cdot 1^{16} i^{6} : 74613 i^{6}

Simplify

\frac{22 !}{7 ! ( 22 - 7 ) !} \cdot 1^{15} i^{7} : 170544 i^{7}

Simplify

\frac{22 !}{8 ! ( 22 - 8 ) !} \cdot 1^{14} i^{8} : 319770 i^{8}

Simplify

\frac{22 !}{9 ! ( 22 - 9 ) !} \cdot 1^{13} i^{9} : 497420 i^{9}

Simplify

\frac{22 !}{10 ! ( 22 - 10 ) !} \cdot 1^{12} i^{10} : 646646 i^{10}

Simplify

\frac{22 !}{11 ! ( 22 - 11 ) !} \cdot 1^{11} i^{11} : 705432 i^{11}

Simplify

\frac{22 !}{12 ! ( 22 - 12 ) !} \cdot 1^{10} i^{12} : 646646 i^{12}

Simplify

\frac{22 !}{13 ! ( 22 - 13 ) !} \cdot 1^{9} i^{13} : 497420 i^{13}

Simplify

\frac{22 !}{14 ! ( 22 - 14 ) !} \cdot 1^{8} i^{14} : 319770 i^{14}

Simplify

\frac{22 !}{15 ! ( 22 - 15 ) !} \cdot 1^{7} i^{15} : 170544 i^{15}

Simplify

\frac{22 !}{16 ! ( 22 - 16 ) !} \cdot 1^{6} i^{16} : 74613 i^{16}

Simplify

\frac{22 !}{17 ! ( 22 - 17 ) !} \cdot 1^{5} i^{17} : 26334 i^{17}

Simplify

\frac{22 !}{18 ! ( 22 - 18 ) !} \cdot 1^{4} i^{18} : 7315 i^{18}

Simplify

\frac{22 !}{19 ! ( 22 - 19 ) !} \cdot 1^{3} i^{19} : 1540 i^{19}

Simplify

\frac{22 !}{20 ! ( 22 - 20 ) !} \cdot 1^{2} i^{20} : 231 i^{20}

Simplify

\frac{22 !}{21 ! ( 22 - 21 ) !} \cdot 1^{1} i^{21} : 22 i^{21}

Simplify

\frac{22 !}{22 ! ( 22 - 22 ) !} \cdot 1^{0} i^{22} : i^{22}

= 1 + 22 i + 231 i^{2} + 1540 i^{3} + 7315 i^{4} + 26334 i^{5} + 74613 i^{6} + 170544 i^{7} + 319770 i^{8} + 497420 i^{9} + 646646 i^{10} + 705432 i^{11} + 646646 i^{12} + 497420 i^{13} + 319770 i^{14} + 170544 i^{15} + 74613 i^{16} + 26334 i^{17} + 7315 i^{18} + 1540 i^{19} + 231 i^{20} + 22 i^{21} + i^{22}

Simplify

1 + 22 i + 231 i^{2} + 1540 i^{3} + 7315 i^{4} + 26334 i^{5} + 74613 i^{6} + 170544 i^{7} + 319770 i^{8} + 497420 i^{9} + 646646 i^{10} + 705432 i^{11} + 646646 i^{12} + 497420 i^{13} + 319770 i^{14} + 170544 i^{15} + 74613 i^{16} + 26334 i^{17} + 7315 i^{18} + 1540 i^{19} + 231 i^{20} + 22 i^{21} + i^{22} : - 2048 i

= - 2048 i

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Frequently Asked Questions (FAQ)

What is (1+i)^{22} ?
The solution to (1+i)^{22} is -2048i

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