Question

Find the value of
i) w¹⁸
ii) w²¹
iii) w⁻³⁰
iv) w⁻¹⁰⁵

Answer 1

Solution:

To find the values of given expression,we first know that what are complex cube roots of unity

$1 + w + {w}^{2} = 0 \\ \\ {w}^{3} = 1$
$1) {w}^{18} = {( {w}^{3}) }^{6} \\ \\ since \: \: {w}^{3} = 1 \\ \\ so \\ \\ {( {w}^{3}) }^{6} = {(1)}^{6} \\ \\ {w}^{18} = 1$
$2) {w}^{21} = {( {w}^{3}) }^{7} \\ \\ = {(1)}^{7} \\ \\ {w}^{21}= 1$
$3) {w}^{ - 30} = \frac{1}{ {w}^{30} } \\ \\ = \frac{1}{ { ({w}^{3}) }^{10} } \\ \\ = \frac{1}{ ({1})^{10} } \\ \\ = \frac{1}{1} \\ \\ {w}^{ - 30}= 1$
$4){w}^{ - 105} = \frac{1}{ {w}^{105} } \\ \\ = \frac{1}{ { ({w}^{3}) }^{35} } \\ \\ = \frac{1}{ ({1})^{35} } \\ \\ = \frac{1}{1} \\ \\ {w}^{ - 105}= 1$
Hope it helps you.

Answer 2

Answer:

here well all know that what is answer