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Question Number 119138 by 675480065 last updated on 22/Oct/20

$$\sqrt{2\sqrt{4\sqrt{8\sqrt{16\sqrt{32...}}}}}=?$$$$\mathrm{pls}\mathrm{help}$$

Commented by Dwaipayan Shikari last updated on 22/Oct/20

$$4=\sqrt{2.8}$$$$4=\sqrt{2\sqrt{64}}$$$$4=\sqrt{2\sqrt{4\sqrt{256}}}$$$$4=\sqrt{2\sqrt{4\sqrt{8\sqrt{1024}}}}$$$$4=\sqrt{2\sqrt{4\sqrt{8\sqrt{16\sqrt{32\sqrt{{32}^2.{64}^2...}...}}}}}$$$$\sqrt{2\sqrt{4\sqrt{8\sqrt{16\sqrt{32...}}}}}=4$$$$$$

Answered by PRITHWISH SEN 2 last updated on 22/Oct/20

$${\mathrm{n}}^2=\sqrt{\mathrm{n}.{\mathrm{n}}^3}=\sqrt{\mathrm{n}.\sqrt{{\mathrm{n}}^6}}=\sqrt{\mathrm{n}.\sqrt{{\mathrm{n}}^2{\mathrm{n}}^4}}=\sqrt{\mathrm{n}.\sqrt{{\mathrm{n}}^2.\sqrt{{\mathrm{n}}^{16}}}}$$$$=\sqrt{\mathrm{n}.\sqrt{{\mathrm{n}}^2.\sqrt{{\mathrm{n}}^3\sqrt{{\mathrm{n}}^4.\sqrt{{\mathrm{n}}^5...}}}}}$$$$\mathrm{n}=2$$$$4=\sqrt{2\sqrt{4\sqrt{8\sqrt{16.....}}}}$$$$\mathrm{n}=3$$$$9=\sqrt{3\sqrt{9\sqrt{27\sqrt{81.....}}}}$$$$.........$$

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