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Question Number 82881 by jagoll last updated on 25/Feb/20

$$$$$$\sqrt{\sqrt{...\sqrt{6561}}}=3^{8^{\mathrm{x}}}\left(60\mathrm{times}\right)$$$$\mathrm{find}\mathrm{x}$$

Commented by mr W last updated on 25/Feb/20

$${6561}^{{\left(\frac{1}{2}\right)}^{60}}=3^{8^x}$$$$3^{8\times \frac{1}{8^{20}}}=3^{8^x}$$$$3^{8^{-19}}=3^{8^x}$$$$\Rightarrow x=-19$$

Commented by jagoll last updated on 25/Feb/20

$$\mathrm{thank}\mathrm{you}$$

Answered by john santu last updated on 25/Feb/20

$$\sqrt{\sqrt{...\sqrt{81}}}=3^{8^{\mathrm{x}}}\left(59\mathrm{times}\right)$$$$\sqrt{\sqrt{...\sqrt{9}}}=3^{8^{\mathrm{x}}}\left(58\mathrm{times}\right)$$$$\sqrt{\sqrt{...\sqrt{3}}}=3^{8^{\mathrm{x}}}\left(57\mathrm{times}\right)$$$${(...{\left(3^{\frac{1}{2}}\right)}^{\frac{1}{2}}...)}^{\frac{1}{2}}=3^{8^{\mathrm{x}}}$$$$\Rightarrow \left(3^{{\left(\frac{1}{2}\right)}^{57}}\right)=3^{8^{\mathrm{x}}}$$$$\Rightarrow 2^{-57}=8^{\mathrm{x}}$$$$\Rightarrow 2^{-57}=2^{3\mathrm{x}}\Rightarrow -57=3\mathrm{x}$$$$\therefore \mathrm{x}=-19$$

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