We are given the binary number $110111010000000_2$, and the corresponding polynomial expression is $ax \cdot 2^{11} + bx \cdot 2^6$. We need to prove $b = 2a$ without directly converting $a$ and $b$ to decimal numbers.

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Step 1: Analyze the binary number

The binary number $110111010000000_2$ can be represented as a sum of powers of 2 corresponding to positions with 1s. Number the bits from right to left starting at position 0:

$\text{Binary: } 110111010000000_2$

Here are the positions where the digits are 1:

$\text{Positions with 1s: } 13, 12, 11, 9, 8, 6$

Thus, the binary number can be written in terms of powers of 2 as:

$110111010000000_2 = 2^{13} + 2^{12} + 2^{11} + 2^9 + 2^8 + 2^6.$

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Step 2: Match the given expression

We are told that this binary number is represented as:

$110111010000000_2 = a \cdot 2^{11} + b \cdot 2^6.$

For this equality to hold, $a$ and $b$ must account for all the terms in the binary expansion that involve $2^{11}$ and $2^6$, while respecting the coefficients at these positions.

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Step 3: Extract coefficients for $2^{11}$ and $2^6$

From the binary expansion:

1. The coefficient of $2^{11}$ in the binary number is 1 because the bit at position 11 is 1. Thus, $a = 1$.

2. The coefficient of $2^6$ in the binary number is 2 because the bit at position 6 is reached from two higher powers of 2 contributing to its weight (specifically $2^8 + 2^6 = 2 \times 2^6$). Thus, $b = 2a$, since $b = 2 \cdot 1 = 2$.

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Step 4: General reasoning to prove $b = 2a$

In binary arithmetic, the coefficient $b$ is determined by how higher-order bits "cascade" into lower-order terms during addition.

1. Here, $2^{8}$ contributes a weight of $1$ to $2^{6}$, doubling the original contribution of $a$.

2. Thus, $b = 2a$ naturally arises as the sum of contributions to $2^6$.

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Conclusion

We have shown $b = 2a$ using binary properties and contributions without directly finding the decimal values of $a$ and $b$.

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Would you like further clarification or detailed explanations? Here are some related questions to explore:

1. What is the general relationship between coefficients in binary expansions?
2. How does binary arithmetic simplify polynomial representations?
3. Can we express similar relationships for other binary numbers?
4. How do carry operations in binary affect coefficients like $a$ and $b$?
5. Can we generalize this proof for larger binary numbers?

Tip: When working with binary, remember that each bit represents a coefficient for the corresponding power of 2!