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Popular >

simplify (1+i)^{18}

Solution

simplify

(1 + i)^{18}

Solution

512 i

Solution steps

(1 + i)^{18}

Apply binomial theorem:

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

a = 1, b = i

= i = 0 \sum 18 (i 18) \cdot 1^{(18 - i)} i^{i}

Expand summation

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

Simplify

\frac{18 !}{0 ! ( 18 - 0 ) !} \cdot 1^{18} i^{0} : 1

Simplify

\frac{18 !}{1 ! ( 18 - 1 ) !} \cdot 1^{17} i^{1} : 18 i

Simplify

\frac{18 !}{2 ! ( 18 - 2 ) !} \cdot 1^{16} i^{2} : 153 i^{2}

Simplify

\frac{18 !}{3 ! ( 18 - 3 ) !} \cdot 1^{15} i^{3} : 816 i^{3}

Simplify

\frac{18 !}{4 ! ( 18 - 4 ) !} \cdot 1^{14} i^{4} : 3060 i^{4}

Simplify

\frac{18 !}{5 ! ( 18 - 5 ) !} \cdot 1^{13} i^{5} : 8568 i^{5}

Simplify

\frac{18 !}{6 ! ( 18 - 6 ) !} \cdot 1^{12} i^{6} : 18564 i^{6}

Simplify

\frac{18 !}{7 ! ( 18 - 7 ) !} \cdot 1^{11} i^{7} : 31824 i^{7}

Simplify

\frac{18 !}{8 ! ( 18 - 8 ) !} \cdot 1^{10} i^{8} : 43758 i^{8}

Simplify

\frac{18 !}{9 ! ( 18 - 9 ) !} \cdot 1^{9} i^{9} : 48620 i^{9}

Simplify

\frac{18 !}{10 ! ( 18 - 10 ) !} \cdot 1^{8} i^{10} : 43758 i^{10}

Simplify

\frac{18 !}{11 ! ( 18 - 11 ) !} \cdot 1^{7} i^{11} : 31824 i^{11}

Simplify

\frac{18 !}{12 ! ( 18 - 12 ) !} \cdot 1^{6} i^{12} : 18564 i^{12}

Simplify

\frac{18 !}{13 ! ( 18 - 13 ) !} \cdot 1^{5} i^{13} : 8568 i^{13}

Simplify

\frac{18 !}{14 ! ( 18 - 14 ) !} \cdot 1^{4} i^{14} : 3060 i^{14}

Simplify

\frac{18 !}{15 ! ( 18 - 15 ) !} \cdot 1^{3} i^{15} : 816 i^{15}

Simplify

\frac{18 !}{16 ! ( 18 - 16 ) !} \cdot 1^{2} i^{16} : 153 i^{16}

Simplify

\frac{18 !}{17 ! ( 18 - 17 ) !} \cdot 1^{1} i^{17} : 18 i^{17}

Simplify

\frac{18 !}{18 ! ( 18 - 18 ) !} \cdot 1^{0} i^{18} : i^{18}

= 1 + 18 i + 153 i^{2} + 816 i^{3} + 3060 i^{4} + 8568 i^{5} + 18564 i^{6} + 31824 i^{7} + 43758 i^{8} + 48620 i^{9} + 43758 i^{10} + 31824 i^{11} + 18564 i^{12} + 8568 i^{13} + 3060 i^{14} + 816 i^{15} + 153 i^{16} + 18 i^{17} + i^{18}

Simplify

1 + 18 i + 153 i^{2} + 816 i^{3} + 3060 i^{4} + 8568 i^{5} + 18564 i^{6} + 31824 i^{7} + 43758 i^{8} + 48620 i^{9} + 43758 i^{10} + 31824 i^{11} + 18564 i^{12} + 8568 i^{13} + 3060 i^{14} + 816 i^{15} + 153 i^{16} + 18 i^{17} + i^{18} : 512 i

= 512 i

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Frequently Asked Questions (FAQ)

What is (1+i)^{18} ?
The solution to (1+i)^{18} is 512i