Convert the binary number 10000000000 (base 2) to its decimal (base 10) value. Show the power-of-two reasoning.

Introduction / Context:
Recognizing powers of two in binary numbers is vital for quick conversions and for understanding address alignment, buffer sizes, and bit masks. A single 1 followed by n zeros in binary equals 2^n in decimal.

Given Data / Assumptions:

• Binary value: 10000000000 (one 1 followed by ten zeros).
• Base is explicitly 2.
• We seek the base-10 value.

Concept / Approach:
If the only set bit is at position n (counting the rightmost bit as position 0), then the decimal value is 2^n. Here, there are ten zeros to the right of the leading 1, so n = 10. Therefore, the decimal value is 2^10 = 1024.

Step-by-Step Solution:
Count zeros to the right of the leftmost 1: there are 10.Compute 2^10 using powers of two: 2^10 = 1024.State decimal result: 1024.Choose option “1024.”

Verification / Alternative check:
Place-value expansion: 12^10 + 02^9 + … + 0*2^0 = 1024 + 0 + … + 0 = 1024, confirming the same result.

Why Other Options Are Wrong:
1034, 1124, and 1037 are unrelated to powers of two; they would correspond to different binary patterns. “None” is false because 1024 is correct.

Common Pitfalls:
Miscounting zeros (off-by-one errors) and forgetting that the rightmost position is 2^0. Always count positions carefully from the least significant bit.

Final Answer:
1024