Question

If w is the complex cube root of unity, find the value of
i) $w+\frac{1}{w}$
ii) w² + w³ + w⁴
iii) (1 + w²)³

Answer 1

Solution:

Properties of complex cube roots of unity

$1 +w + {w}^{2} = 0 \\ \\ {w}^{3} = 1$
To find the values of given expression,use above discussed properties

$1) \: w + \frac{1}{w} \\ \\ = \frac{ {w}^{2} + 1}{w} \\ \\ = \frac{ - w}{w} \\ \\ = - 1 \\ \\ so \\ \\ \: w + \frac{1}{w} = - 1$
$2) {w}^{2} + {w}^{3} + {w}^{4} \\ \\ = {w}^{2} (1 + w + {w}^{2} ) \\ \\ = {w}^{2} (1) \\ \\ {w}^{2} + {w}^{3} + {w}^{4} = {w}^{2}$
By the same way

$3) {(1 + {w}^{2} )}^{3} = {( - w)}^{3} \\ \\ = {( - 1)}^{3} ( {w})^{3} \\ \\ = - 1(1) \\ \\ = - 1 \\ \\ so \\ \\ {(1 + {w}^{2} )}^{3} = - 1$
Hope it helps you.