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Let w be a complex number such that w^3=1 Find all possible values of 1/(1+w) + 1/(1+w^2).

 Mar 9, 2020
 #1
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The possible value of 1/(1 + w) and 1/(1 + w^2) are 1 and 2.

 Mar 9, 2020
 #2
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Let \(w\) be a complex number such that \(w^3=1\) Find all possible values of \(\dfrac{1}{1+w} + \dfrac{1}{1+w^2}\).

 

\(w^3 = ww^2 ~\text{ or }~ w^2=\dfrac{w^3}{w} \)

 

\(\begin{array}{|rcll|} \hline && \mathbf{\dfrac{1}{1+w} + \dfrac{1}{1+w^2}} \quad & | \quad w^2=\dfrac{w^3}{w} \\\\ &=& \dfrac{1}{1+w} + \dfrac{1}{1+\dfrac{w^3}{w}} \\\\ &=& \dfrac{1}{1+w} + \dfrac{w}{w+w^3} \quad & | \quad w^3=1 \\\\ &=& \dfrac{1}{1+w} + \dfrac{w}{w+1} \\\\ &=& \dfrac{1+w}{1+w} \\\\ &=& \mathbf{1} \\ \hline \end{array}\)

 

laugh

 Mar 9, 2020

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