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Question Number 162102 by HongKing last updated on 26/Dec/21

$$\mathrm{Solve}\mathrm{for}\mathrm{real}\mathrm{numbers}:$$$$5^{\mathbf{x}}+4^{\frac{1}{\mathbf{x}}}+{25}^{\mathbf{x}}\cdot {16}^{\frac{1}{\mathbf{x}}}=2527$$

Commented by mr W last updated on 26/Dec/21

$$suchthat{4}^{\frac{1}{x}}isinteger,x=1or2$$$$trywithx=1:$$$$5^1+4^1+{25}^1\times {16}^1=409$$$$trywithx=2:$$$$5^2+4^{\frac{1}{2}}+{25}^2\times {16}^{\frac{1}{2}}=2527✓$$$$\Rightarrow x=2istheroot$$

Commented by HongKing last updated on 28/Dec/21

$$\mathrm{thank}\mathrm{you}\mathrm{dear}\mathrm{Sir}$$

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