Question:
If $\omega$ is an imaginary cube root of unity, then (1 + $\omega$ - $\omega^2)^7$ is equal to
If $\omega$ is an imaginary cube root of unity, then (1 + $\omega$ - $\omega^2)^7$ is equal to
Updated On: Jun 14, 2022
- 128$\omega$
- -128$\omega$
- 128$\omega^2$
- -128$\omega^2$
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The Correct Option is D
Solution and Explanation
$(1+\omega-\omega^2)^7=(-\omega^2-\omega^2)^7 \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, [\because 1+\omega+\omega^2=0] $
$=(-2\omega^2)^7=(-2)^7\omega^{14}=-128 \omega^2$
$=(-2\omega^2)^7=(-2)^7\omega^{14}=-128 \omega^2$
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Concepts Used:
Complex Numbers and Quadratic Equations
Complex Number: Any number that is formed as a+ib is called a complex number. For example: 9+3i,7+8i are complex numbers. Here i = -1. With this we can say that i² = 1. So, for every equation which does not have a real solution we can use i = -1.
Quadratic equation: A polynomial that has two roots or is of the degree 2 is called a quadratic equation. The general form of a quadratic equation is y=ax²+bx+c. Here a≠0, b and c are the real numbers.
