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Popular >

simplify (-1+i)^{15}

Solution

simplify

(- 1 + i)^{15}

Solution

- 128 - 128 i

Solution steps

(- 1 + i)^{15}

Apply binomial theorem:

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

a = - 1, b = i

= i = 0 \sum 15 (i 15) (- 1)^{(15 - i)} i^{i}

Expand summation

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

Simplify

\frac{15 !}{0 ! ( 15 - 0 ) !} (- 1)^{15} i^{0} : - 1

Simplify

\frac{15 !}{1 ! ( 15 - 1 ) !} (- 1)^{14} i^{1} : 15 i

Simplify

\frac{15 !}{2 ! ( 15 - 2 ) !} (- 1)^{13} i^{2} : - 105 i^{2}

Simplify

\frac{15 !}{3 ! ( 15 - 3 ) !} (- 1)^{12} i^{3} : 455 i^{3}

Simplify

\frac{15 !}{4 ! ( 15 - 4 ) !} (- 1)^{11} i^{4} : - 1365 i^{4}

Simplify

\frac{15 !}{5 ! ( 15 - 5 ) !} (- 1)^{10} i^{5} : 3003 i^{5}

Simplify

\frac{15 !}{6 ! ( 15 - 6 ) !} (- 1)^{9} i^{6} : - 5005 i^{6}

Simplify

\frac{15 !}{7 ! ( 15 - 7 ) !} (- 1)^{8} i^{7} : 6435 i^{7}

Simplify

\frac{15 !}{8 ! ( 15 - 8 ) !} (- 1)^{7} i^{8} : - 6435 i^{8}

Simplify

\frac{15 !}{9 ! ( 15 - 9 ) !} (- 1)^{6} i^{9} : 5005 i^{9}

Simplify

\frac{15 !}{10 ! ( 15 - 10 ) !} (- 1)^{5} i^{10} : - 3003 i^{10}

Simplify

\frac{15 !}{11 ! ( 15 - 11 ) !} (- 1)^{4} i^{11} : 1365 i^{11}

Simplify

\frac{15 !}{12 ! ( 15 - 12 ) !} (- 1)^{3} i^{12} : - 455 i^{12}

Simplify

\frac{15 !}{13 ! ( 15 - 13 ) !} (- 1)^{2} i^{13} : 105 i^{13}

Simplify

\frac{15 !}{14 ! ( 15 - 14 ) !} (- 1)^{1} i^{14} : - 15 i^{14}

Simplify

\frac{15 !}{15 ! ( 15 - 15 ) !} (- 1)^{0} i^{15} : i^{15}

= - 1 + 15 i - 105 i^{2} + 455 i^{3} - 1365 i^{4} + 3003 i^{5} - 5005 i^{6} + 6435 i^{7} - 6435 i^{8} + 5005 i^{9} - 3003 i^{10} + 1365 i^{11} - 455 i^{12} + 105 i^{13} - 15 i^{14} + i^{15}

Simplify

- 1 + 15 i - 105 i^{2} + 455 i^{3} - 1365 i^{4} + 3003 i^{5} - 5005 i^{6} + 6435 i^{7} - 6435 i^{8} + 5005 i^{9} - 3003 i^{10} + 1365 i^{11} - 455 i^{12} + 105 i^{13} - 15 i^{14} + i^{15} : - 128 i - 128

= - 128 i - 128

Rewrite in standard complex form:

- 128 - 128 i

= - 128 - 128 i

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Frequently Asked Questions (FAQ)

What is (-1+i)^{15} ?
The solution to (-1+i)^{15} is -128-128i