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Popular >

simplify (1+i)^{21}

Solution

simplify

(1 + i)^{21}

Solution

- 1024 - 1024 i

Solution steps

(1 + i)^{21}

Apply binomial theorem:

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

a = 1, b = i

= i = 0 \sum 21 (i 21) \cdot 1^{(21 - i)} i^{i}

Expand summation

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

Simplify

\frac{21 !}{0 ! ( 21 - 0 ) !} \cdot 1^{21} i^{0} : 1

Simplify

\frac{21 !}{1 ! ( 21 - 1 ) !} \cdot 1^{20} i^{1} : 21 i

Simplify

\frac{21 !}{2 ! ( 21 - 2 ) !} \cdot 1^{19} i^{2} : 210 i^{2}

Simplify

\frac{21 !}{3 ! ( 21 - 3 ) !} \cdot 1^{18} i^{3} : 1330 i^{3}

Simplify

\frac{21 !}{4 ! ( 21 - 4 ) !} \cdot 1^{17} i^{4} : 5985 i^{4}

Simplify

\frac{21 !}{5 ! ( 21 - 5 ) !} \cdot 1^{16} i^{5} : 20349 i^{5}

Simplify

\frac{21 !}{6 ! ( 21 - 6 ) !} \cdot 1^{15} i^{6} : 54264 i^{6}

Simplify

\frac{21 !}{7 ! ( 21 - 7 ) !} \cdot 1^{14} i^{7} : 116280 i^{7}

Simplify

\frac{21 !}{8 ! ( 21 - 8 ) !} \cdot 1^{13} i^{8} : 203490 i^{8}

Simplify

\frac{21 !}{9 ! ( 21 - 9 ) !} \cdot 1^{12} i^{9} : 293930 i^{9}

Simplify

\frac{21 !}{10 ! ( 21 - 10 ) !} \cdot 1^{11} i^{10} : 352716 i^{10}

Simplify

\frac{21 !}{11 ! ( 21 - 11 ) !} \cdot 1^{10} i^{11} : 352716 i^{11}

Simplify

\frac{21 !}{12 ! ( 21 - 12 ) !} \cdot 1^{9} i^{12} : 293930 i^{12}

Simplify

\frac{21 !}{13 ! ( 21 - 13 ) !} \cdot 1^{8} i^{13} : 203490 i^{13}

Simplify

\frac{21 !}{14 ! ( 21 - 14 ) !} \cdot 1^{7} i^{14} : 116280 i^{14}

Simplify

\frac{21 !}{15 ! ( 21 - 15 ) !} \cdot 1^{6} i^{15} : 54264 i^{15}

Simplify

\frac{21 !}{16 ! ( 21 - 16 ) !} \cdot 1^{5} i^{16} : 20349 i^{16}

Simplify

\frac{21 !}{17 ! ( 21 - 17 ) !} \cdot 1^{4} i^{17} : 5985 i^{17}

Simplify

\frac{21 !}{18 ! ( 21 - 18 ) !} \cdot 1^{3} i^{18} : 1330 i^{18}

Simplify

\frac{21 !}{19 ! ( 21 - 19 ) !} \cdot 1^{2} i^{19} : 210 i^{19}

Simplify

\frac{21 !}{20 ! ( 21 - 20 ) !} \cdot 1^{1} i^{20} : 21 i^{20}

Simplify

\frac{21 !}{21 ! ( 21 - 21 ) !} \cdot 1^{0} i^{21} : i^{21}

= 1 + 21 i + 210 i^{2} + 1330 i^{3} + 5985 i^{4} + 20349 i^{5} + 54264 i^{6} + 116280 i^{7} + 203490 i^{8} + 293930 i^{9} + 352716 i^{10} + 352716 i^{11} + 293930 i^{12} + 203490 i^{13} + 116280 i^{14} + 54264 i^{15} + 20349 i^{16} + 5985 i^{17} + 1330 i^{18} + 210 i^{19} + 21 i^{20} + i^{21}

Simplify

1 + 21 i + 210 i^{2} + 1330 i^{3} + 5985 i^{4} + 20349 i^{5} + 54264 i^{6} + 116280 i^{7} + 203490 i^{8} + 293930 i^{9} + 352716 i^{10} + 352716 i^{11} + 293930 i^{12} + 203490 i^{13} + 116280 i^{14} + 54264 i^{15} + 20349 i^{16} + 5985 i^{17} + 1330 i^{18} + 210 i^{19} + 21 i^{20} + i^{21} : - 1024 i - 1024

= - 1024 i - 1024

Rewrite in standard complex form:

- 1024 - 1024 i

= - 1024 - 1024 i

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Frequently Asked Questions (FAQ)

What is (1+i)^{21} ?
The solution to (1+i)^{21} is -1024-1024i