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Determine the value of the infinite product \((2^{1/3})(4^{1/9})(8^{1/27})(16^{1/81}) \dotsm\) Enter your answer in the form "\sqrt[a]{b}", which stands for \(\sqrt[a]{b}\)

 Feb 28, 2021
 #1
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Hello!

 

I love your pfp :))

So, lets turn everything where the base is 2. 

2^(1/3) is already in base 2.

4^(1/9) = (2^2)^(1/9) = 2^(2/9)

8^(1/27) = (2^3)^(1/27) = 2^(3/27) = 2^(1/9)

16^(1/81) = (2^4)^(1/81) = 2^(4/81)

Now that everything is in base 2, we can just add. :))

 

1/3+2/9+1/9+4/81 = 58/81

 

2^(58/81) = <81>sqrt(2^58)

 

I hope this helped. :))

 

=^._.^=

 Feb 28, 2021
 #2
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productfor(n, 1, 1000,  (2^n)^(3^-n))=1.6817928305 074290861==2^(3/4)==(2^3)^(1/4)

 Feb 28, 2021

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