To solve the given question:

The table appears to show a representation of numbers using place values. Specifically, it looks like the columns under "Place Value" are powers of a base, and the row below labeled "Number" represents coefficients for those place values.

Part A: Determine the base for the place value table.

The table provides the following place values: 128, ____, 16, ____, 1. These numbers are powers of a certain base.

• 128 = $2^7$
• 16 = $2^4$
• 1 = $2^0$

Clearly, these numbers are powers of 2. This means the base of the number system in this table is 2.

Conclusion for Part A:

The base of the place value system is 2.

Would you like to explore more details on the next parts of the question or further clarifications?

Here are 5 additional related questions to deepen your understanding:

1. What are place values in other common bases, like base-10 or base-16?
2. How would the representation change if the base were 8 or 10?
3. How can you convert a binary number (base-2) to a decimal number (base-10)?
4. What is the significance of powers in positional number systems?
5. How does place value affect the weight of a digit in any given number system?

Tip: When working with place value tables, always remember that the base indicates the number of unique digits available and how place values scale as powers of the base.