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Inequality

If w ≠ 1 is...

Question

If $w \neq 1$ is a cube root of unity, and $a + b = 11, a^{3} + b^{3} = 1001$ , find the value of $(a w^{2} + b w) (a w + b w^{2})$ .

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Solution

given that $a + b = 11$ & $a^{3} + b^{3} = 1001$
Since, $w$ is the cube root of unity.
Then, $w^{3} = 1$ & $1 + w + w^{2} = 0$
now, $y = (a w^{2} + b w) (a w + b w^{2}) = a^{2} + b^{2} + a b (w + w^{2}) = a^{2} + b^{2} - a b$
$\Rightarrow y = \frac{a^{3} + b^{3}}{a + b} = \frac{1001}{11} = 91$
Ans: 91

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