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

$$Findallvaluesofx:$$$${\left(2^x\right)}^{x^2-8}=32$$

Answered by MJS last updated on 26/Nov/19

$${\left(a^b\right)}^c=a^{bc}$$$$2^{x^3-8x}=2^5$$$$x^3-8x-5=0$$$$\mathrm{this}\mathrm{has}3\mathrm{real}\mathrm{solutions}\Rightarrow \mathrm{we}\mathrm{need}\mathrm{the}$$$$\mathrm{trigonometric}\mathrm{formula}$$$$x_1=\frac{4\sqrt{6}}{3}\sin (\frac{\pi }{3}+\frac{1}{3}\arcsin \frac{15\sqrt{6}}{64})\approx 3.10043$$$$x_2=-\frac{4\sqrt{6}}{3}\sin (\frac{\pi }{6}+\frac{1}{3}\arcsin \frac{15\sqrt{6}}{64})\approx -2.43931$$$$x_3=-\frac{4\sqrt{6}}{3}\sin \left(\frac{1}{3}\arcsin \frac{15\sqrt{6}}{64}\right)\approx -.661120$$

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