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Popular >

simplify (-1+i)^6

Solution

simplify

(- 1 + i)^{6}

Solution

8 i

Solution steps

(- 1 + i)^{6}

Apply binomial theorem:

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

a = - 1, b = i

= i = 0 \sum 6 (i 6) (- 1)^{(6 - i)} i^{i}

Expand summation

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

Simplify

\frac{6 !}{0 ! ( 6 - 0 ) !} (- 1)^{6} i^{0} : 1

Simplify

\frac{6 !}{1 ! ( 6 - 1 ) !} (- 1)^{5} i^{1} : - 6 i

Simplify

\frac{6 !}{2 ! ( 6 - 2 ) !} (- 1)^{4} i^{2} : 15 i^{2}

Simplify

\frac{6 !}{3 ! ( 6 - 3 ) !} (- 1)^{3} i^{3} : - 20 i^{3}

Simplify

\frac{6 !}{4 ! ( 6 - 4 ) !} (- 1)^{2} i^{4} : 15 i^{4}

Simplify

\frac{6 !}{5 ! ( 6 - 5 ) !} (- 1)^{1} i^{5} : - 6 i^{5}

Simplify

\frac{6 !}{6 ! ( 6 - 6 ) !} (- 1)^{0} i^{6} : i^{6}

= 1 - 6 i + 15 i^{2} - 20 i^{3} + 15 i^{4} - 6 i^{5} + i^{6}

Simplify

1 - 6 i + 15 i^{2} - 20 i^{3} + 15 i^{4} - 6 i^{5} + i^{6} : 8 i

= 8 i

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54 x^{3} - 135 x^{2} + 93 x - 18

factor 4n^2-4 factor

4 n^{2} - 4

6 x^{2} + 7 x - 98 = 0

\frac{4}{3} x + 2 = \frac{x}{2} + 1

Frequently Asked Questions (FAQ)

What is (-1+i)^6 ?
The solution to (-1+i)^6 is 8i