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Question Number 72190 by oyemi kemewari last updated on 26/Oct/19

$$3^{\mathrm{x}}+4^{\mathrm{x}}=5^{\mathrm{x}}$$

Commented by oyemi kemewari last updated on 26/Oct/19

find x

Commented by Joel578 last updated on 26/Oct/19

$$x=2$$

Answered by mind is power last updated on 27/Oct/19

$$\iff {\left(\frac{3}{5}\right)}^{\mathrm{x}}+{\left(\frac{4}{5}\right)}^{\mathrm{x}}=1$$$$\mathrm{x}=2\mathrm{worcks}$$$$\mathrm{f}\left(\mathrm{x}\right)={\left(\frac{3}{5}\right)}^{\mathrm{x}}+{\left(\frac{4}{5}\right)}^{\mathrm{x}}$$$${\mathrm{f}}^{\prime }\left(\mathrm{x}\right)=\ln \left(\frac{3}{5}\right).{\left(\frac{3}{5}\right)}^{\mathrm{x}}+\ln \left(\frac{4}{5}\right).{\left(\frac{4}{5}\right)}^{\mathrm{x}}<0$$$$\mathrm{f}\left(2\right)=1\Rightarrow \mathrm{has}\mathrm{only}2\mathrm{as}\mathrm{solution}$$

Commented by Rasheed.Sindhi last updated on 27/Oct/19

$${\left(\frac{3}{5}\right)}^{\mathrm{x}}+{\left(\frac{4}{5}\right)}^{\mathrm{x}}=1$$$$Assume\frac{3}{5}=\sin \theta \Rightarrow \frac{4}{5}=\cos \theta$$$${\left(\sin \theta \right)}^x+{\left(\cos \theta \right)}^x=1$$$$Comparingwiththeidentity$$$${\sin }^2\theta +{\cos }^2\theta =1,wehave$$$$x=2$$

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