# If (zÂ²+1/zÂ²) =18, find the value of zÂ³-1/zÂ³, using only the positive value of z-1/z..

We know that,

$(z- \frac{1}{z})^{2} = z^{2} + \frac{1}{z^{2}} - 2\times z \times \frac{1}{z} \\ = z^{2} + \frac{1}{z^{2}} - 2 = 18 - 2 = 16 \\ z -\frac{1}{z} = +4, - 4$

taking positive value and cube both side,

$(z -\frac{1}{z})^{3} = 64\\ z^{3} - \frac{1}{z^3} - 3 \times z \times \frac{1}{z} \times (z-\frac{1}{z}) = 64 \\ z^{3} - \frac{1}{z^{3}} = 64 + 12 = 76\\$

Hence,

$z^{3} - \frac{1}{z^{3}} = 64 + 12 = 76\\$

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