### Instruction: Find $2^{\frac{1}{2}} \cdot 4^{\frac{1}{4}} \cdot 8^{\frac{1}{8}} \cdot 16^{\frac{1}{16}} \dotsm.$.### Response: This is an infinite product of powers of 2, where the exponent of each term is half the exponent of the previous term.
I can rewrite each term as $2^{\frac{1}{2}} \cdot 2^{\frac{1}{2} \cdot \frac{1}{2}} \cdot 2^{\frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2}} \dotsm,$ using the rule that $(a^b)^c = a^{bc}.$
Now I can combine the exponents of 2, since they are all multiplied: $2^{\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{2} + \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} + \dotsm}.$
I notice that the exponents form a geometric series, where the first term is $\frac{1}{2}$ and the common ratio is $\frac{1}{2}.$
I can use the formula for the sum of an infinite geometric series, which is $\frac{a}{1-r}$ where $a$ is the first term and $r$ is the common ratio.
Plugging in the values, I get $\frac{\frac{1}{2}}{1-\frac{1}{2}} = 1.$
Therefore, the exponent of 2 is 1, and the product is $2^1 = 2.$