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Popular >

simplify (-1+i)^9

Solution

simplify

(- 1 + i)^{9}

Solution

- 16 + 16 i

Solution steps

(- 1 + i)^{9}

Apply binomial theorem:

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

a = - 1, b = i

= i = 0 \sum 9 (i 9) (- 1)^{(9 - i)} i^{i}

Expand summation

= \frac{9 !}{0 ! ( 9 - 0 ) !} (- 1)^{9} i^{0} + \frac{9 !}{1 ! ( 9 - 1 ) !} (- 1)^{8} i^{1} + \frac{9 !}{2 ! ( 9 - 2 ) !} (- 1)^{7} i^{2} + \frac{9 !}{3 ! ( 9 - 3 ) !} (- 1)^{6} i^{3} + \frac{9 !}{4 ! ( 9 - 4 ) !} (- 1)^{5} i^{4} + \frac{9 !}{5 ! ( 9 - 5 ) !} (- 1)^{4} i^{5} + \frac{9 !}{6 ! ( 9 - 6 ) !} (- 1)^{3} i^{6} + \frac{9 !}{7 ! ( 9 - 7 ) !} (- 1)^{2} i^{7} + \frac{9 !}{8 ! ( 9 - 8 ) !} (- 1)^{1} i^{8} + \frac{9 !}{9 ! ( 9 - 9 ) !} (- 1)^{0} i^{9}

Simplify

\frac{9 !}{0 ! ( 9 - 0 ) !} (- 1)^{9} i^{0} : - 1

Simplify

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

Simplify

\frac{9 !}{2 ! ( 9 - 2 ) !} (- 1)^{7} i^{2} : - 36 i^{2}

Simplify

\frac{9 !}{3 ! ( 9 - 3 ) !} (- 1)^{6} i^{3} : 84 i^{3}

Simplify

\frac{9 !}{4 ! ( 9 - 4 ) !} (- 1)^{5} i^{4} : - 126 i^{4}

Simplify

\frac{9 !}{5 ! ( 9 - 5 ) !} (- 1)^{4} i^{5} : 126 i^{5}

Simplify

\frac{9 !}{6 ! ( 9 - 6 ) !} (- 1)^{3} i^{6} : - 84 i^{6}

Simplify

\frac{9 !}{7 ! ( 9 - 7 ) !} (- 1)^{2} i^{7} : 36 i^{7}

Simplify

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

Simplify

\frac{9 !}{9 ! ( 9 - 9 ) !} (- 1)^{0} i^{9} : i^{9}

= - 1 + 9 i - 36 i^{2} + 84 i^{3} - 126 i^{4} + 126 i^{5} - 84 i^{6} + 36 i^{7} - 9 i^{8} + i^{9}

Simplify

- 1 + 9 i - 36 i^{2} + 84 i^{3} - 126 i^{4} + 126 i^{5} - 84 i^{6} + 36 i^{7} - 9 i^{8} + i^{9} : 16 i - 16

= 16 i - 16

Rewrite in standard complex form:

- 16 + 16 i

= - 16 + 16 i

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Frequently Asked Questions (FAQ)

What is (-1+i)^9 ?
The solution to (-1+i)^9 is -16+16i