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# algebra

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The product

$2^{1/4} \cdot 2^{1/8} \cdot 2^{1/16} \cdot 2^{1/32} \dotsb$

can be expressed in the form $\sqrt[a]{b}$, where $a$ and $b$ are positive integers.  Find the smallest possible value of $a + b$.

Mar 29, 2021

#1
+22096
+2

Multiplying numbers with the same base (for this problem, the base is 2) can be simplified by adding the powers.

1/4  +  1/8  +  1/16  +  1/32  ...  =  1/2

The fractions form an infinite geometric series whose first term is  1/4  and whose common ratio is  1/2.

Sum =  (1/4)  /  [ 1 - 1/2 ]    =  (1/4)  /  (1/2)  =  1/2

So, the product can be written as  21/2  =  sqrt( 2 )

This means that  a = 2  and  b = 2.

a + b  =  2 + 2  =  4

Mar 29, 2021

#1
+22096
+2

Multiplying numbers with the same base (for this problem, the base is 2) can be simplified by adding the powers.

1/4  +  1/8  +  1/16  +  1/32  ...  =  1/2

The fractions form an infinite geometric series whose first term is  1/4  and whose common ratio is  1/2.

Sum =  (1/4)  /  [ 1 - 1/2 ]    =  (1/4)  /  (1/2)  =  1/2

So, the product can be written as  21/2  =  sqrt( 2 )

This means that  a = 2  and  b = 2.

a + b  =  2 + 2  =  4

geno3141 Mar 29, 2021