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Popular >

simplify (1+i)^{17}

Solution

simplify

(1 + i)^{17}

Solution

256 + 256 i

Solution steps

(1 + i)^{17}

Apply binomial theorem:

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

a = 1, b = i

= i = 0 \sum 17 (i 17) \cdot 1^{(17 - i)} i^{i}

Expand summation

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

Simplify

\frac{17 !}{0 ! ( 17 - 0 ) !} \cdot 1^{17} i^{0} : 1

Simplify

\frac{17 !}{1 ! ( 17 - 1 ) !} \cdot 1^{16} i^{1} : 17 i

Simplify

\frac{17 !}{2 ! ( 17 - 2 ) !} \cdot 1^{15} i^{2} : 136 i^{2}

Simplify

\frac{17 !}{3 ! ( 17 - 3 ) !} \cdot 1^{14} i^{3} : 680 i^{3}

Simplify

\frac{17 !}{4 ! ( 17 - 4 ) !} \cdot 1^{13} i^{4} : 2380 i^{4}

Simplify

\frac{17 !}{5 ! ( 17 - 5 ) !} \cdot 1^{12} i^{5} : 6188 i^{5}

Simplify

\frac{17 !}{6 ! ( 17 - 6 ) !} \cdot 1^{11} i^{6} : 12376 i^{6}

Simplify

\frac{17 !}{7 ! ( 17 - 7 ) !} \cdot 1^{10} i^{7} : 19448 i^{7}

Simplify

\frac{17 !}{8 ! ( 17 - 8 ) !} \cdot 1^{9} i^{8} : 24310 i^{8}

Simplify

\frac{17 !}{9 ! ( 17 - 9 ) !} \cdot 1^{8} i^{9} : 24310 i^{9}

Simplify

\frac{17 !}{10 ! ( 17 - 10 ) !} \cdot 1^{7} i^{10} : 19448 i^{10}

Simplify

\frac{17 !}{11 ! ( 17 - 11 ) !} \cdot 1^{6} i^{11} : 12376 i^{11}

Simplify

\frac{17 !}{12 ! ( 17 - 12 ) !} \cdot 1^{5} i^{12} : 6188 i^{12}

Simplify

\frac{17 !}{13 ! ( 17 - 13 ) !} \cdot 1^{4} i^{13} : 2380 i^{13}

Simplify

\frac{17 !}{14 ! ( 17 - 14 ) !} \cdot 1^{3} i^{14} : 680 i^{14}

Simplify

\frac{17 !}{15 ! ( 17 - 15 ) !} \cdot 1^{2} i^{15} : 136 i^{15}

Simplify

\frac{17 !}{16 ! ( 17 - 16 ) !} \cdot 1^{1} i^{16} : 17 i^{16}

Simplify

\frac{17 !}{17 ! ( 17 - 17 ) !} \cdot 1^{0} i^{17} : i^{17}

= 1 + 17 i + 136 i^{2} + 680 i^{3} + 2380 i^{4} + 6188 i^{5} + 12376 i^{6} + 19448 i^{7} + 24310 i^{8} + 24310 i^{9} + 19448 i^{10} + 12376 i^{11} + 6188 i^{12} + 2380 i^{13} + 680 i^{14} + 136 i^{15} + 17 i^{16} + i^{17}

Simplify

1 + 17 i + 136 i^{2} + 680 i^{3} + 2380 i^{4} + 6188 i^{5} + 12376 i^{6} + 19448 i^{7} + 24310 i^{8} + 24310 i^{9} + 19448 i^{10} + 12376 i^{11} + 6188 i^{12} + 2380 i^{13} + 680 i^{14} + 136 i^{15} + 17 i^{16} + i^{17} : 256 i + 256

= 256 i + 256

Rewrite in standard complex form:

256 + 256 i

= 256 + 256 i

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Frequently Asked Questions (FAQ)

What is (1+i)^{17} ?
The solution to (1+i)^{17} is 256+256i