If a and b are real numbers such that
${\left( {2 + \alpha } \right)^4} = a + b\alpha$
where $\alpha = {{ - 1 + i\sqrt 3 } \over 2}$ then a + b is
equal to :
Solution
$\alpha = \omega$ as given $\alpha = {{ - 1 + i\sqrt 3 } \over 2}$
<br><br>$\Rightarrow {(2 + \omega )^4} = a + b\omega \,({\omega ^3} = 1)$<br><br>$$ \Rightarrow {2^4} + {4.2^3}\omega + {6.2^2}{\omega ^3} + 4.2.\,{\omega ^3} + {\omega ^4} = a + b\omega $$<br><br>$\Rightarrow 16 + 32\omega + 24{\omega ^2} + 8 + \omega = a + b\omega$<br><br>$\Rightarrow 24 + 24{\omega ^2} + 33\omega = a + b\omega$<br><br><br>$\Rightarrow - 24\omega + 33\omega = a + b\omega$<br><br>$\Rightarrow a = 0,\,b = 9$
About this question
Subject: Mathematics · Chapter: Complex Numbers and Quadratic Equations · Topic: Complex Numbers and Argand Plane
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