What is the value of #2sqrt(2sqrt(2sqrt(2)))#?

3 Answers
George C.
Aug 28, 2016

#2sqrt(2sqrt(2sqrt(2)))= root(8)(2^15)= 2^(15/8)~~ 3.6680161728#

Explanation:

#2sqrt(2sqrt(2sqrt(2)))#

#= sqrt(2^2*2sqrt(2sqrt(2)))#

#= sqrt(2^3sqrt(2sqrt(2)))#

#= sqrt(sqrt(2^6*2sqrt(2)))#

#= sqrt(sqrt(2^7sqrt(2)))#

#= sqrt(sqrt(sqrt(2^14*2)))#

#= sqrt(sqrt(sqrt(2^15)))#

#= root(8)(2^15)#

#= 2^(15/8)#

#~~ 3.6680161728#

Cesareo R.
Aug 28, 2016

#3.66802#

Explanation:

#2sqrt(2sqrt(2sqrt(2))) =2^1 2^(1/2)2^(1/4)2^(1/8) = 2^(1+1/2+1/4+1/8) = 2 xx 2^(7/8) = 3.66802#

Ratnaker Mehta
Aug 28, 2016

#2^(15/8)~=3.6680161728186849269741663791766#.

Explanation:

#2sqrt(2sqrt(2(sqrt2)))#

#=2sqrt(2sqrt(2*2^(1/2)))#

#=2sqrt(2sqrt(2^(1+1/2)#

#=2sqrt(2sqrt(2^(3/2)))#

#=2sqrt(2*(2^(3/2))^(1/2)#

#=2sqrt(2*2^((3/2)*(1/2))#

#=2sqrt(2*2^(3/4))#

#=2sqrt(2^(1+3/4))#

#=2sqrt(2^(7/4))#

#=2*2^((7/4)*(1/2))#

#=2*2^(7/8)#

#=2^(1+7/8)#

#=2^(15/8)#.

Its numerical value, using scientific calculator :

#2^(15/8)~=3.6680161728186849269741663791766#.

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