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  3. If (z²+1/z²) =18, find the value of z³-1/z³, using only the positive value of z-1/z..
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If (z²+1/z²) =18, find the value of z³-1/z³, using only the positive value of z-1/z..

Answers (1)
D Deependra Verma

We know that,

                         $$ (z- frac1z)^2 = z^2 + frac1z^2 - 2 imes z imes frac1z $$\$$ $$ = z^2 + frac1z^2 - 2 = 18 - 2 = 16 $$\$$ $$ z -frac1z = +4, - 4

taking positive value and cube both side, 

                                             $$ (z -frac1z)^3 = 64$$\$$ $$ z^3 - frac1z^3 - 3 imes z imes frac1z imes (z-frac1z) = 64 $$\$$ $$ z^3 - frac1z^3 = 64 + 12 = 76$$\$$

Hence, 

              $$ z^3 - frac1z^3 = 64 + 12 = 76$$\$$

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