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Compute \(\large\sqrt{9\sqrt{27\sqrt{128\sqrt{9\sqrt{27\sqrt{128\sqrt{...}}}}}}} \)

 Jul 26, 2020
 #1
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Let x =  sqrt 9 sqrt27 sqrt128.......

 

square it      square it     square it again

 

and you have     x8 =    612220032 sqrt 9 sqrt27sqrt128.....    the red part is 'x'   

(61222032 is 94272128)

                           x=61222032 x

                           x^7 = 612220032

                            x = 18                          ????

 Jul 26, 2020
edited by ElectricPavlov  Jul 26, 2020
edited by ElectricPavlov  Jul 26, 2020
edited by ElectricPavlov  Jul 26, 2020
edited by ElectricPavlov  Jul 26, 2020
edited by ElectricPavlov  Jul 26, 2020
 #2
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Find the Prime Factors of each:

 

1 - 9 = 3^2   and   27 =3^3    and    128  =2^7

 

2 - By taking an infinite number of Square Roots, The exponents will eventually converge to 1 for each Prime Factor.

 

3 - Then, will be left with the factors themselves multiplied by each other.

 

4 - Therefore: 3  x  3  x  2 = 18

 Jul 26, 2020

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