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

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

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simplify (1/2+(sqrt(3))/2 i)^3

simplify

(\frac{1}{2} + \frac{3}{2} i)^{3}

- 1

Solution steps

(\frac{1}{2} + \frac{3}{2} i)^{3}

Apply Perfect Cube Formula:

(a + b)^{3} = a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}

(\frac{1}{2} + \frac{3}{2} i)^{3} = (\frac{1}{2})^{3} + 3 (\frac{1}{2})^{2} \frac{3}{2} i + 3 \cdot \frac{1}{2} (\frac{3}{2} i)^{2} + (\frac{3}{2} i)^{3}

= (\frac{1}{2})^{3} + 3 (\frac{1}{2})^{2} \frac{3}{2} i + 3 \cdot \frac{1}{2} (\frac{3}{2} i)^{2} + (\frac{3}{2} i)^{3}

Simplify

(\frac{1}{2})^{3} + 3 (\frac{1}{2})^{2} \frac{3}{2} i + 3 \cdot \frac{1}{2} (\frac{3}{2} i)^{2} + (\frac{3}{2} i)^{3} : - 1

= - 1

(x^{2} + 2) (x^{2} - 2)

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