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 $a + b$.
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
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