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Popular Algebra >

simplify (1+i)^{28}

Solution

simplify

(1 + i)^{28}

Solution

- 16384

Solution steps

(1 + i)^{28}

Apply binomial theorem:

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

a = 1, b = i

= i = 0 \sum 28 (i 28) \cdot 1^{(28 - i)} i^{i}

Expand summation

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

Simplify

\frac{28 !}{0 ! ( 28 - 0 ) !} \cdot 1^{28} i^{0} : 1

Simplify

\frac{28 !}{1 ! ( 28 - 1 ) !} \cdot 1^{27} i^{1} : 28 i

Simplify

\frac{28 !}{2 ! ( 28 - 2 ) !} \cdot 1^{26} i^{2} : 378 i^{2}

Simplify

\frac{28 !}{3 ! ( 28 - 3 ) !} \cdot 1^{25} i^{3} : 3276 i^{3}

Simplify

\frac{28 !}{4 ! ( 28 - 4 ) !} \cdot 1^{24} i^{4} : 20475 i^{4}

Simplify

\frac{28 !}{5 ! ( 28 - 5 ) !} \cdot 1^{23} i^{5} : 98280 i^{5}

Simplify

\frac{28 !}{6 ! ( 28 - 6 ) !} \cdot 1^{22} i^{6} : 376740 i^{6}

Simplify

\frac{28 !}{7 ! ( 28 - 7 ) !} \cdot 1^{21} i^{7} : 1184040 i^{7}

Simplify

\frac{28 !}{8 ! ( 28 - 8 ) !} \cdot 1^{20} i^{8} : 3108105 i^{8}

Simplify

\frac{28 !}{9 ! ( 28 - 9 ) !} \cdot 1^{19} i^{9} : 6906900 i^{9}

Simplify

\frac{28 !}{10 ! ( 28 - 10 ) !} \cdot 1^{18} i^{10} : 13123110 i^{10}

Simplify

\frac{28 !}{11 ! ( 28 - 11 ) !} \cdot 1^{17} i^{11} : 21474180 i^{11}

Simplify

\frac{28 !}{12 ! ( 28 - 12 ) !} \cdot 1^{16} i^{12} : 30421755 i^{12}

Simplify

\frac{28 !}{13 ! ( 28 - 13 ) !} \cdot 1^{15} i^{13} : 37442160 i^{13}

Simplify

\frac{28 !}{14 ! ( 28 - 14 ) !} \cdot 1^{14} i^{14} : 40116600 i^{14}

Simplify

\frac{28 !}{15 ! ( 28 - 15 ) !} \cdot 1^{13} i^{15} : 37442160 i^{15}

Simplify

\frac{28 !}{16 ! ( 28 - 16 ) !} \cdot 1^{12} i^{16} : 30421755 i^{16}

Simplify

\frac{28 !}{17 ! ( 28 - 17 ) !} \cdot 1^{11} i^{17} : 21474180 i^{17}

Simplify

\frac{28 !}{18 ! ( 28 - 18 ) !} \cdot 1^{10} i^{18} : 13123110 i^{18}

Simplify

\frac{28 !}{19 ! ( 28 - 19 ) !} \cdot 1^{9} i^{19} : 6906900 i^{19}

Simplify

\frac{28 !}{20 ! ( 28 - 20 ) !} \cdot 1^{8} i^{20} : 3108105 i^{20}

Simplify

\frac{28 !}{21 ! ( 28 - 21 ) !} \cdot 1^{7} i^{21} : 1184040 i^{21}

Simplify

\frac{28 !}{22 ! ( 28 - 22 ) !} \cdot 1^{6} i^{22} : 376740 i^{22}

Simplify

\frac{28 !}{23 ! ( 28 - 23 ) !} \cdot 1^{5} i^{23} : 98280 i^{23}

Simplify

\frac{28 !}{24 ! ( 28 - 24 ) !} \cdot 1^{4} i^{24} : 20475 i^{24}

Simplify

\frac{28 !}{25 ! ( 28 - 25 ) !} \cdot 1^{3} i^{25} : 3276 i^{25}

Simplify

\frac{28 !}{26 ! ( 28 - 26 ) !} \cdot 1^{2} i^{26} : 378 i^{26}

Simplify

\frac{28 !}{27 ! ( 28 - 27 ) !} \cdot 1^{1} i^{27} : 28 i^{27}

Simplify

\frac{28 !}{28 ! ( 28 - 28 ) !} \cdot 1^{0} i^{28} : i^{28}

= 1 + 28 i + 378 i^{2} + 3276 i^{3} + 20475 i^{4} + 98280 i^{5} + 376740 i^{6} + 1184040 i^{7} + 3108105 i^{8} + 6906900 i^{9} + 13123110 i^{10} + 21474180 i^{11} + 30421755 i^{12} + 37442160 i^{13} + 40116600 i^{14} + 37442160 i^{15} + 30421755 i^{16} + 21474180 i^{17} + 13123110 i^{18} + 6906900 i^{19} + 3108105 i^{20} + 1184040 i^{21} + 376740 i^{22} + 98280 i^{23} + 20475 i^{24} + 3276 i^{25} + 378 i^{26} + 28 i^{27} + i^{28}

Simplify

1 + 28 i + 378 i^{2} + 3276 i^{3} + 20475 i^{4} + 98280 i^{5} + 376740 i^{6} + 1184040 i^{7} + 3108105 i^{8} + 6906900 i^{9} + 13123110 i^{10} + 21474180 i^{11} + 30421755 i^{12} + 37442160 i^{13} + 40116600 i^{14} + 37442160 i^{15} + 30421755 i^{16} + 21474180 i^{17} + 13123110 i^{18} + 6906900 i^{19} + 3108105 i^{20} + 1184040 i^{21} + 376740 i^{22} + 98280 i^{23} + 20475 i^{24} + 3276 i^{25} + 378 i^{26} + 28 i^{27} + i^{28} : - 16384

= - 16384

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Frequently Asked Questions (FAQ)

What is (1+i)^{28} ?
The solution to (1+i)^{28} is -16384

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