What is (-1/2-(sqrt(3))/2 i)^4 ?

The solution to (-1/2-(sqrt(3))/2 i)^4 is -1/2-(sqrt(3))/2 i

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simplify (-1/2-(sqrt(3))/2 i)^4

simplify

(- \frac{1}{2} - \frac{3}{2} i)^{4}

- \frac{1}{2} - \frac{3}{2} i

Solution steps

(- \frac{1}{2} - \frac{3}{2} i)^{4}

Apply binomial theorem:

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

a = - \frac{1}{2}, b = - \frac{3}{2} i

= i = 0 \sum 4 (i 4) (- \frac{1}{2})^{(4 - i)} (- \frac{3}{2} i)^{i}

Expand summation

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

Simplify

\frac{4 !}{0 ! ( 4 - 0 ) !} (- \frac{1}{2})^{4} (- \frac{3}{2} i)^{0} : \frac{1}{16}

Simplify

\frac{4 !}{1 ! ( 4 - 1 ) !} (- \frac{1}{2})^{3} (- \frac{3}{2} i)^{1} : \frac{3 i}{4}

Simplify

\frac{4 !}{2 ! ( 4 - 2 ) !} (- \frac{1}{2})^{2} (- \frac{3}{2} i)^{2} : \frac{9 i ^{2}}{8}

Simplify

\frac{4 !}{3 ! ( 4 - 3 ) !} (- \frac{1}{2})^{1} (- \frac{3}{2} i)^{3} : \frac{3 3 i ^{3}}{4}

Simplify

\frac{4 !}{4 ! ( 4 - 4 ) !} (- \frac{1}{2})^{0} (- \frac{3}{2} i)^{4} : \frac{9 i ^{4}}{16}

= \frac{1}{16} + \frac{3 i}{4} + \frac{9 i ^{2}}{8} + \frac{3 3 i ^{3}}{4} + \frac{9 i ^{4}}{16}

Simplify

\frac{1}{16} + \frac{3 i}{4} + \frac{9 i ^{2}}{8} + \frac{3 3 i ^{3}}{4} + \frac{9 i ^{4}}{16} : - \frac{1}{2} - \frac{3}{2} i

= - \frac{1}{2} - \frac{3}{2} i

- 0.2 x = 1.6

x^{3} + 8 x + x - 42

∣ x + 5 ∣ = ∣ x - 3 ∣

∣ x - 8 ∣ = - 3

x^{2} + 200 x + 0.000015 = 0

What is (-1/2-(sqrt(3))/2 i)^4 ?
The solution to (-1/2-(sqrt(3))/2 i)^4 is -1/2-(sqrt(3))/2 i