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Determine the value of the infinite product

\((2^{1/3})(2^{1/9})(2^{1/27}) \dotsm\)

  Enter your answer in the form "\sqrt[a]{b}"

 May 14, 2022
 #1
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Suppose you mean the exponents form a geometric sequence.

Just combine the exponents using the formula \(a^m a^n = a^{m+n}\).

 

\(2^{1/3} \cdot 2^{1/9} \cdot 2^{1/27} \cdots = 2^{1/3 + 1/9 + 1/27 + \cdots}\)

 

Note that the exponent is now the sum of a geometric sequence. Can you complete it now?

 May 14, 2022

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