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Popular >

simplify (-1+i)^{10}

Solution

simplify

(- 1 + i)^{10}

Solution

- 32 i

Solution steps

(- 1 + i)^{10}

Apply binomial theorem:

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

a = - 1, b = i

= i = 0 \sum 10 (i 10) (- 1)^{(10 - i)} i^{i}

Expand summation

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

Simplify

\frac{10 !}{0 ! ( 10 - 0 ) !} (- 1)^{10} i^{0} : 1

Simplify

\frac{10 !}{1 ! ( 10 - 1 ) !} (- 1)^{9} i^{1} : - 10 i

Simplify

\frac{10 !}{2 ! ( 10 - 2 ) !} (- 1)^{8} i^{2} : 45 i^{2}

Simplify

\frac{10 !}{3 ! ( 10 - 3 ) !} (- 1)^{7} i^{3} : - 120 i^{3}

Simplify

\frac{10 !}{4 ! ( 10 - 4 ) !} (- 1)^{6} i^{4} : 210 i^{4}

Simplify

\frac{10 !}{5 ! ( 10 - 5 ) !} (- 1)^{5} i^{5} : - 252 i^{5}

Simplify

\frac{10 !}{6 ! ( 10 - 6 ) !} (- 1)^{4} i^{6} : 210 i^{6}

Simplify

\frac{10 !}{7 ! ( 10 - 7 ) !} (- 1)^{3} i^{7} : - 120 i^{7}

Simplify

\frac{10 !}{8 ! ( 10 - 8 ) !} (- 1)^{2} i^{8} : 45 i^{8}

Simplify

\frac{10 !}{9 ! ( 10 - 9 ) !} (- 1)^{1} i^{9} : - 10 i^{9}

Simplify

\frac{10 !}{10 ! ( 10 - 10 ) !} (- 1)^{0} i^{10} : i^{10}

= 1 - 10 i + 45 i^{2} - 120 i^{3} + 210 i^{4} - 252 i^{5} + 210 i^{6} - 120 i^{7} + 45 i^{8} - 10 i^{9} + i^{10}

Simplify

1 - 10 i + 45 i^{2} - 120 i^{3} + 210 i^{4} - 252 i^{5} + 210 i^{6} - 120 i^{7} + 45 i^{8} - 10 i^{9} + i^{10} : - 32 i

= - 32 i

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Frequently Asked Questions (FAQ)

What is (-1+i)^{10} ?
The solution to (-1+i)^{10} is -32i