6z-4-z-z-2-1-z-2-z-4-

Question Number 149770 by mathdanisur last updated on 07/Aug/21

6 z^{4} = z^{z^{2}} \Rightarrow 1 + z^{2} + z^{4} = ?

Commented by amin96 last updated on 07/Aug/21

z^{2} = {l o g}_{z} 6 z^{4} \Rightarrow z^{2} = {l o g}_{z} 6 + 4

z^{2} = {l o g}_{6} z {l o g}_{6} z = {l o g}_{z} 6 + 4

{l o g}_{6} z - \frac{1}{{l o g}_{6} z} = 4 {l o g}_{6} z = x z > 1

x^{2} - 4 x - 1 = 0 x = \frac{4 + 2 \sqrt{3}}{2} = 2 + \sqrt{3}

{l o g}_{6} z = 2 + \sqrt{3} z^{2} = 2 + \sqrt{3} z^{4} = 7 + 4 \sqrt{3}

1 + z^{2} + z^{4} = 1 + 2 + \sqrt{3} + 7 + 4 \sqrt{3} = 10 + 5 \sqrt{3}

Commented by mathdanisur last updated on 07/Aug/21

Thanks S er