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Popular Algebra >

simplify (1+i)^{16}

Solution

simplify

(1 + i)^{16}

Solution

256

Solution steps

(1 + i)^{16}

Apply binomial theorem:

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

a = 1, b = i

= i = 0 \sum 16 (i 16) \cdot 1^{(16 - i)} i^{i}

Expand summation

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

Simplify

\frac{16 !}{0 ! ( 16 - 0 ) !} \cdot 1^{16} i^{0} : 1

Simplify

\frac{16 !}{1 ! ( 16 - 1 ) !} \cdot 1^{15} i^{1} : 16 i

Simplify

\frac{16 !}{2 ! ( 16 - 2 ) !} \cdot 1^{14} i^{2} : 120 i^{2}

Simplify

\frac{16 !}{3 ! ( 16 - 3 ) !} \cdot 1^{13} i^{3} : 560 i^{3}

Simplify

\frac{16 !}{4 ! ( 16 - 4 ) !} \cdot 1^{12} i^{4} : 1820 i^{4}

Simplify

\frac{16 !}{5 ! ( 16 - 5 ) !} \cdot 1^{11} i^{5} : 4368 i^{5}

Simplify

\frac{16 !}{6 ! ( 16 - 6 ) !} \cdot 1^{10} i^{6} : 8008 i^{6}

Simplify

\frac{16 !}{7 ! ( 16 - 7 ) !} \cdot 1^{9} i^{7} : 11440 i^{7}

Simplify

\frac{16 !}{8 ! ( 16 - 8 ) !} \cdot 1^{8} i^{8} : 12870 i^{8}

Simplify

\frac{16 !}{9 ! ( 16 - 9 ) !} \cdot 1^{7} i^{9} : 11440 i^{9}

Simplify

\frac{16 !}{10 ! ( 16 - 10 ) !} \cdot 1^{6} i^{10} : 8008 i^{10}

Simplify

\frac{16 !}{11 ! ( 16 - 11 ) !} \cdot 1^{5} i^{11} : 4368 i^{11}

Simplify

\frac{16 !}{12 ! ( 16 - 12 ) !} \cdot 1^{4} i^{12} : 1820 i^{12}

Simplify

\frac{16 !}{13 ! ( 16 - 13 ) !} \cdot 1^{3} i^{13} : 560 i^{13}

Simplify

\frac{16 !}{14 ! ( 16 - 14 ) !} \cdot 1^{2} i^{14} : 120 i^{14}

Simplify

\frac{16 !}{15 ! ( 16 - 15 ) !} \cdot 1^{1} i^{15} : 16 i^{15}

Simplify

\frac{16 !}{16 ! ( 16 - 16 ) !} \cdot 1^{0} i^{16} : i^{16}

= 1 + 16 i + 120 i^{2} + 560 i^{3} + 1820 i^{4} + 4368 i^{5} + 8008 i^{6} + 11440 i^{7} + 12870 i^{8} + 11440 i^{9} + 8008 i^{10} + 4368 i^{11} + 1820 i^{12} + 560 i^{13} + 120 i^{14} + 16 i^{15} + i^{16}

Simplify

1 + 16 i + 120 i^{2} + 560 i^{3} + 1820 i^{4} + 4368 i^{5} + 8008 i^{6} + 11440 i^{7} + 12870 i^{8} + 11440 i^{9} + 8008 i^{10} + 4368 i^{11} + 1820 i^{12} + 560 i^{13} + 120 i^{14} + 16 i^{15} + i^{16} : 256

= 256

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Frequently Asked Questions (FAQ)

What is (1+i)^{16} ?
The solution to (1+i)^{16} is 256

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