To determine which expressions represent your score after 12 hits in Game 2, where the score starts at 2 and doubles every time, let's analyze the problem step by step.
### Problem Breakdown:
1. Initial Score: The initial score is 2.
2. Doubling Every Time: The score doubles after each hit.
### Calculating the Score After 12 Hits:
The score starts at 2 and doubles every hit. We can represent this as multiplying the score by 2 for every hit. Thus, the score after 12 hits can be represented as:
[tex]\[ 2 \times 2 \times 2 \times \cdots \times 2 \text{ (12 times)} \][/tex]
This is the same as:
[tex]\[ 2^{12} \][/tex]
Let's convert the problem into a more understandable form using these 4 options and see if they match our calculation:
1. Sum of 2 added 12 times:
[tex]\[ \frac{2 + 2 + \cdots + 2}{12 \text { times }} \][/tex]
This expression represents the sum of 2 added 12 times. Mathematically, it is:
[tex]\[ 2 + 2 + 2 + \cdots + 2 \text { (12 times)} = 2 \times 12 = 24 \][/tex]
This is not what we are looking for because it is a linear increase, not an exponential doubling.
2. Product of 2 multiplied 12 times:
[tex]\[ \frac{2 \cdot 2 \cdot 2 \cdot \ldots \cdot 2}{12 \text { times }} \][/tex]
This expression shows 2 being multiplied by itself 12 times, which is:
[tex]\[ 2 \cdot 2 \cdot 2 \cdot \ldots \cdot 2 = 2^{12} \][/tex]
This matches our requirement of exponential growth. Indeed, this is a correct representation.
3. 2 multiplied by 12:
[tex]\[ 2 \cdot 12 \][/tex]
This is simply 2 multiplied by 12. Mathematically, it results in:
[tex]\[ 2 \times 12 = 24 \][/tex]
This, again, is a linear relationship and doesn’t represent the exponential doubling we need.
4. Exponential expression:
[tex]\[ 2^{12} \][/tex]
This clearly represents 2 multiplied by itself 12 times. It directly translates to the exponential growth we are interested in:
[tex]\[ 2^{12} \][/tex]
This is a correct representation.
### Conclusion:
The expressions that correctly represent the score after 12 hits, where the score doubles every time starting from 2, are:
[tex]\[
\boxed{2^{12}}
\][/tex]
and
[tex]\[
\boxed{2 \cdot 2 \cdot 2 \cdot \ldots \cdot 2 \text{ (12 times)}}
\][/tex]
Both of these expressions accurately reflect the exponential growth of the score in the game.
