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Popular Algebra >

simplify (1+i)^{25}

Solution

simplify

(1 + i)^{25}

Solution

4096 + 4096 i

Solution steps

(1 + i)^{25}

Apply binomial theorem:

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

a = 1, b = i

= i = 0 \sum 25 (i 25) \cdot 1^{(25 - i)} i^{i}

Expand summation

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

Simplify

\frac{25 !}{0 ! ( 25 - 0 ) !} \cdot 1^{25} i^{0} : 1

Simplify

\frac{25 !}{1 ! ( 25 - 1 ) !} \cdot 1^{24} i^{1} : 25 i

Simplify

\frac{25 !}{2 ! ( 25 - 2 ) !} \cdot 1^{23} i^{2} : 300 i^{2}

Simplify

\frac{25 !}{3 ! ( 25 - 3 ) !} \cdot 1^{22} i^{3} : 2300 i^{3}

Simplify

\frac{25 !}{4 ! ( 25 - 4 ) !} \cdot 1^{21} i^{4} : 12650 i^{4}

Simplify

\frac{25 !}{5 ! ( 25 - 5 ) !} \cdot 1^{20} i^{5} : 53130 i^{5}

Simplify

\frac{25 !}{6 ! ( 25 - 6 ) !} \cdot 1^{19} i^{6} : 177100 i^{6}

Simplify

\frac{25 !}{7 ! ( 25 - 7 ) !} \cdot 1^{18} i^{7} : 480700 i^{7}

Simplify

\frac{25 !}{8 ! ( 25 - 8 ) !} \cdot 1^{17} i^{8} : 1081575 i^{8}

Simplify

\frac{25 !}{9 ! ( 25 - 9 ) !} \cdot 1^{16} i^{9} : 2042975 i^{9}

Simplify

\frac{25 !}{10 ! ( 25 - 10 ) !} \cdot 1^{15} i^{10} : 3268760 i^{10}

Simplify

\frac{25 !}{11 ! ( 25 - 11 ) !} \cdot 1^{14} i^{11} : 4457400 i^{11}

Simplify

\frac{25 !}{12 ! ( 25 - 12 ) !} \cdot 1^{13} i^{12} : 5200300 i^{12}

Simplify

\frac{25 !}{13 ! ( 25 - 13 ) !} \cdot 1^{12} i^{13} : 5200300 i^{13}

Simplify

\frac{25 !}{14 ! ( 25 - 14 ) !} \cdot 1^{11} i^{14} : 4457400 i^{14}

Simplify

\frac{25 !}{15 ! ( 25 - 15 ) !} \cdot 1^{10} i^{15} : 3268760 i^{15}

Simplify

\frac{25 !}{16 ! ( 25 - 16 ) !} \cdot 1^{9} i^{16} : 2042975 i^{16}

Simplify

\frac{25 !}{17 ! ( 25 - 17 ) !} \cdot 1^{8} i^{17} : 1081575 i^{17}

Simplify

\frac{25 !}{18 ! ( 25 - 18 ) !} \cdot 1^{7} i^{18} : 480700 i^{18}

Simplify

\frac{25 !}{19 ! ( 25 - 19 ) !} \cdot 1^{6} i^{19} : 177100 i^{19}

Simplify

\frac{25 !}{20 ! ( 25 - 20 ) !} \cdot 1^{5} i^{20} : 53130 i^{20}

Simplify

\frac{25 !}{21 ! ( 25 - 21 ) !} \cdot 1^{4} i^{21} : 12650 i^{21}

Simplify

\frac{25 !}{22 ! ( 25 - 22 ) !} \cdot 1^{3} i^{22} : 2300 i^{22}

Simplify

\frac{25 !}{23 ! ( 25 - 23 ) !} \cdot 1^{2} i^{23} : 300 i^{23}

Simplify

\frac{25 !}{24 ! ( 25 - 24 ) !} \cdot 1^{1} i^{24} : 25 i^{24}

Simplify

\frac{25 !}{25 ! ( 25 - 25 ) !} \cdot 1^{0} i^{25} : i^{25}

= 1 + 25 i + 300 i^{2} + 2300 i^{3} + 12650 i^{4} + 53130 i^{5} + 177100 i^{6} + 480700 i^{7} + 1081575 i^{8} + 2042975 i^{9} + 3268760 i^{10} + 4457400 i^{11} + 5200300 i^{12} + 5200300 i^{13} + 4457400 i^{14} + 3268760 i^{15} + 2042975 i^{16} + 1081575 i^{17} + 480700 i^{18} + 177100 i^{19} + 53130 i^{20} + 12650 i^{21} + 2300 i^{22} + 300 i^{23} + 25 i^{24} + i^{25}

Simplify

1 + 25 i + 300 i^{2} + 2300 i^{3} + 12650 i^{4} + 53130 i^{5} + 177100 i^{6} + 480700 i^{7} + 1081575 i^{8} + 2042975 i^{9} + 3268760 i^{10} + 4457400 i^{11} + 5200300 i^{12} + 5200300 i^{13} + 4457400 i^{14} + 3268760 i^{15} + 2042975 i^{16} + 1081575 i^{17} + 480700 i^{18} + 177100 i^{19} + 53130 i^{20} + 12650 i^{21} + 2300 i^{22} + 300 i^{23} + 25 i^{24} + i^{25} : 4096 i + 4096

= 4096 i + 4096

Rewrite in standard complex form:

4096 + 4096 i

= 4096 + 4096 i

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Frequently Asked Questions (FAQ)

What is (1+i)^{25} ?
The solution to (1+i)^{25} is 4096+4096i

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