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Question Number 208519 by Tawa11 last updated on 17/Jun/24

$$\mathrm{If}{\mathrm{a}}^{\mathrm{x}}={\mathrm{b}}^{\mathrm{y}}$$$$\frac{\mathrm{a}}{\mathrm{x}}+\frac{\mathrm{b}}{\mathrm{y}}=1$$$$\mathrm{then},{\mathrm{a}}^{\mathrm{x}}+{\mathrm{b}}^{\mathrm{y}}=?$$

Answered by mr W last updated on 17/Jun/24

$$a^x=b^y=k,say$$$$x=\frac{\log k}{\log a}$$$$y=\frac{\log k}{\log b}$$$$\frac{a\log a}{\log k}+\frac{b\log b}{\log k}=1$$$$a\log a+b\log b=\log a^a{b}^b=\log k$$$$\Rightarrow k=a^a{b}^b$$$$a^x+b^y=2k=2a^a{b}^b✓$$

Commented by Tawa11 last updated on 17/Jun/24

$$\mathrm{I}\mathrm{appreciate}\mathrm{sir}.$$

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