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# square roots

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$\sqrt{2021\sqrt{2021\sqrt{2021\sqrt\cdots}}} = \ ?$

Jan 31, 2021

#2
+113659
+2

\sqrt{2021\sqrt{2021\sqrt{2021\sqrt\cdots}}}

Let

$$y=\sqrt{2021\sqrt{2021\sqrt{2021\sqrt\cdots}}} \\ then\\ y=\sqrt{2021y}\\ y^2=2021y\\ y^2-2021y=0\\ y(y-2021)=0\\ y=0 \;\;or \;\;2021\\ y\ge0\\ so\\ y=2021$$

Jan 31, 2021

#1
0

(44.955 533 59, 301.421 852 9, 780.495 717 2, 1255.938 631, 1593.189 246, 1794.389 998, 1904.327 227, 1961.796 454, 1991.178 202, 2006.033 685, 2013.502 937, 2017.247 986, 2019.123 121, 2020.061 343, 2020.530 617, 2020.765 295, 2020.882 644, 2020.941 321, 2020.970 66, 2020.985 33, 2020.992 665, 2020.996 333, 2020.998 166, 2020.999 083, 2020.999 542, 2020.999 771, 2020.999 885, 2020.999 943, 2020.999 971, 2020.999 986, 2020.999 993, 2020.999 996, 2020.999 998, 2020.999 999, 2021, 2021, 2021, 2021, 2021 ==2021

Jan 31, 2021
#2
+113659
+2

\sqrt{2021\sqrt{2021\sqrt{2021\sqrt\cdots}}}

Let

$$y=\sqrt{2021\sqrt{2021\sqrt{2021\sqrt\cdots}}} \\ then\\ y=\sqrt{2021y}\\ y^2=2021y\\ y^2-2021y=0\\ y(y-2021)=0\\ y=0 \;\;or \;\;2021\\ y\ge0\\ so\\ y=2021$$

Melody Jan 31, 2021
#3
+120023
+1

Nice work, Melody......these nested root  things  always  frustrate me........you made that look  TOO  easy....LOL!!!

CPhill  Jan 31, 2021
#4
+113659
+1

Thanks Chris,

It is the rates questions, that you seem to do effortlesslyy, that always confuse me

Melody  Jan 31, 2021