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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) \cdot 1^{(6 - i)} (- i)^{i}

Expand summation

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

\frac{6 !}{6 ! ( 6 - 6 ) !} \cdot 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

Popular Examples

-1/9 (x-27)+1/3 (x+3)=x+4

- \frac{1}{9} (x - 27) + \frac{1}{3} (x + 3) = x + 4

\frac{x - 5}{x + 2} > 0

factor 3c^2-6c factor

3 c^{2} - 6 c

(x - 7)^{2} - 25 = 0

\frac{m}{3} = 9

Frequently Asked Questions (FAQ)

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