Find-all-values-of-x-2-x-x-2-8-32-

Question Number 74582 by Maclaurin Stickker last updated on 26/Nov/19

F i n d a l l v a l u e s o f x :

{(2^{x})}^{x^{2} - 8} = 32

Answered by MJS last updated on 26/Nov/19

{(a^{b})}^{c} = a^{b c}

2^{x^{3} - 8 x} = 2^{5}

x^{3} - 8 x - 5 = 0

this has 3 real solutions \Rightarrow we need the

trigonometric formula

x_{1} = \frac{4 \sqrt{6}}{3} \sin (\frac{π}{3} + \frac{1}{3} \arcsin \frac{15 \sqrt{6}}{64}) \approx 3.10043

x_{2} = - \frac{4 \sqrt{6}}{3} \sin (\frac{π}{6} + \frac{1}{3} \arcsin \frac{15 \sqrt{6}}{64}) \approx - 2.43931

x_{3} = - \frac{4 \sqrt{6}}{3} \sin (\frac{1}{3} \arcsin \frac{15 \sqrt{6}}{64}) \approx - . 661120

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