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Let  

\(P = 5^{1/5} \cdot 5^{1/25} \cdot 5^{1/125} \dotsm\)

Then P can be expressed in the form a^{b/c}, where a, b, and c are positive integers. Find the smallest possible value of a + b + c.

 Jun 8, 2022
 #1
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Using an exponent  law, we can write P  as

 

5 ^ ( 1/5 + 1/25 + 1/125  .....)

 

The  sum  of the exponents is the sum of an infinite series = 

 

(1/5)  / ( 1 - 1/5)  =  (1/5) / (4/5)   =   1/4

 

So P  can be expressed as

 

5^(1/4)

 

a = 5        b  =   1        c   = 4

 

a + b + c   =   10

 

 

cool cool cool  

 Jun 8, 2022

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