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Popular >

simplify (-1+i)^8

Solution

simplify

(- 1 + i)^{8}

Solution

16

Solution steps

(- 1 + i)^{8}

Apply binomial theorem:

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

a = - 1, b = i

= i = 0 \sum 8 (i 8) (- 1)^{(8 - i)} i^{i}

Expand summation

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

Simplify

\frac{8 !}{0 ! ( 8 - 0 ) !} (- 1)^{8} i^{0} : 1

Simplify

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

Simplify

\frac{8 !}{2 ! ( 8 - 2 ) !} (- 1)^{6} i^{2} : 28 i^{2}

Simplify

\frac{8 !}{3 ! ( 8 - 3 ) !} (- 1)^{5} i^{3} : - 56 i^{3}

Simplify

\frac{8 !}{4 ! ( 8 - 4 ) !} (- 1)^{4} i^{4} : 70 i^{4}

Simplify

\frac{8 !}{5 ! ( 8 - 5 ) !} (- 1)^{3} i^{5} : - 56 i^{5}

Simplify

\frac{8 !}{6 ! ( 8 - 6 ) !} (- 1)^{2} i^{6} : 28 i^{6}

Simplify

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

Simplify

\frac{8 !}{8 ! ( 8 - 8 ) !} (- 1)^{0} i^{8} : i^{8}

= 1 - 8 i + 28 i^{2} - 56 i^{3} + 70 i^{4} - 56 i^{5} + 28 i^{6} - 8 i^{7} + i^{8}

Simplify

1 - 8 i + 28 i^{2} - 56 i^{3} + 70 i^{4} - 56 i^{5} + 28 i^{6} - 8 i^{7} + i^{8} : 16

= 16

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Frequently Asked Questions (FAQ)

What is (-1+i)^8 ?
The solution to (-1+i)^8 is 16