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Popular >

expand (-0.5+0.5i)^5

Solution

expand

(- 0.5 + 0.5 i)^{5}

Solution

0.125 - 0.125 i

Solution steps

(- 0.5 + 0.5 i)^{5}

Apply binomial theorem:

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

a = - 0.5, b = 0.5 i

= i = 0 \sum 5 (i 5) (- 0.5)^{(5 - i)} (0.5 i)^{i}

Expand summation

= \frac{5 !}{0 ! ( 5 - 0 ) !} (- 0.5)^{5} (0.5 i)^{0} + \frac{5 !}{1 ! ( 5 - 1 ) !} (- 0.5)^{4} (0.5 i)^{1} + \frac{5 !}{2 ! ( 5 - 2 ) !} (- 0.5)^{3} (0.5 i)^{2} + \frac{5 !}{3 ! ( 5 - 3 ) !} (- 0.5)^{2} (0.5 i)^{3} + \frac{5 !}{4 ! ( 5 - 4 ) !} (- 0.5)^{1} (0.5 i)^{4} + \frac{5 !}{5 ! ( 5 - 5 ) !} (- 0.5)^{0} (0.5 i)^{5}

\frac{5 !}{0 ! ( 5 - 0 ) !} (- 0.5)^{5} (0.5 i)^{0} = - 0.03125

Simplify

\frac{5 !}{1 ! ( 5 - 1 ) !} (- 0.5)^{4} (0.5 i)^{1} : 0.15625 i

\frac{5 !}{2 ! ( 5 - 2 ) !} (- 0.5)^{3} (0.5 i)^{2} = 0.3125

Simplify

\frac{5 !}{3 ! ( 5 - 3 ) !} (- 0.5)^{2} (0.5 i)^{3} : - 0.3125 i

\frac{5 !}{4 ! ( 5 - 4 ) !} (- 0.5)^{1} (0.5 i)^{4} = - 0.15625

Simplify

\frac{5 !}{5 ! ( 5 - 5 ) !} (- 0.5)^{0} (0.5 i)^{5} : 0.03125 i

= - 0.03125 + 0.15625 i + 0.3125 - 0.3125 i - 0.15625 + 0.03125 i

Simplify

- 0.03125 + 0.15625 i + 0.3125 - 0.3125 i - 0.15625 + 0.03125 i : 0.125 - 0.125 i

= 0.125 - 0.125 i

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Frequently Asked Questions (FAQ)

What is expand (-0.5+0.5i)^5 ?
The solution to expand (-0.5+0.5i)^5 is 0.125-0.125i