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Question Number 145769 by Engr_Jidda last updated on 07/Jul/21

$$findxif{2}^x+2^{3x}=16$$

Commented by 7770 last updated on 08/Jul/21

$$2^{\mathit{x}}+2^{3\mathit{x}}=16$$$$\mathit{l}\mathit{e}\mathit{t}{2}^{\mathit{x}}=\mathit{m}$$$$\mathit{m}+{\mathit{m}}^3=16$$$${\mathit{m}}^3+\mathit{m}-16=0$$$$\mathit{l}\mathit{e}\mathit{t}\mathit{f}\left(\mathit{x}\right)={\mathit{m}}^3+\mathit{m}-16$$$${\mathit{f}}^{{}^{\prime }}\left(\mathit{x}\right)=3{\mathit{m}}^2+1$$$$\mathit{l}\mathit{e}\mathit{t}\mathit{t}\mathit{h}\mathit{e}\mathit{i}\mathit{n}\mathit{i}\mathit{t}\mathit{i}\mathit{a}\mathit{l}\mathit{g}\mathit{u}\mathit{e}\mathit{s}\mathit{s}\mathit{m}\mathit{b}\mathit{e}0$$$$\mathit{u}\mathit{s}\mathit{i}\mathit{n}\mathit{g}\mathit{n}\mathit{r}\mathit{i}$$$${\mathit{m}}_1={\mathit{m}}_0-\frac{\mathit{f}\left(\mathit{x}\right)}{{\mathit{f}}^{{}^{\prime }}\left(\mathit{x}\right)}$$$$\mathit{m}=2.387686553...$$$$2^{\mathit{x}}=\mathit{m}$$$$2^{\mathit{x}}=2.387686553...$$$$\mathit{x}=\frac{\mathit{l}\mathit{n}2.387686553...}{\mathit{l}\mathit{n}2}$$$$\mathit{x}\approx 1.255613$$

Answered by mr W last updated on 07/Jul/21

$$lett=2^x$$$$t^3+t-16=0$$$$\mathrm{\Delta }=\frac{1}{27}+64>0$$$$\sqrt{\mathrm{\Delta }}=\frac{\sqrt{5187}}{9}$$$$\Rightarrow t=\sqrt[3]{\frac{\sqrt{5187}}{9}+8}-\sqrt[3]{\frac{\sqrt{5187}}{9}-8}$$$$\Rightarrow x={\log }_2(\sqrt[3]{\frac{\sqrt{5187}}{9}+8}-\sqrt[3]{\frac{\sqrt{5187}}{9}-8})$$$$\approx 1.255613$$

Commented by Engr_Jidda last updated on 08/Jul/21

$$thankyousir$$

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