Find the square-root of $1100100_{\text {two }}$
Answer (2)
The square root of the binary number 110010 0 two is 101 0 two , after converting it to base-10 and calculating the square root. The base-10 equivalent of 110010 0 two is 100, and the square root of 100 is 10 which converts back to 101 0 two .
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Convert 110010 0 two to base-10: 110010 0 two = 10 0 ten .
Calculate the square root: 100 = 10 .
Convert 10 back to base-2: 1 0 ten = 101 0 two .
The square root of 110010 0 two is 101 0 two .
Explanation
Problem Analysis We are asked to find the square root of 110010 0 two . This means we need to convert the number from base-2 to base-10, find the square root, and then convert the result back to base-2.
Converting to Base-10 First, convert the base-2 number 110010 0 two to base-10. We have:
110010 0 two = 1 × 2 6 + 1 × 2 5 + 0 × 2 4 + 0 × 2 3 + 1 × 2 2 + 0 × 2 1 + 0 × 2 0
110010 0 two = 1 × 64 + 1 × 32 + 0 × 16 + 0 × 8 + 1 × 4 + 0 × 2 + 0 × 1
110010 0 two = 64 + 32 + 4 = 10 0 ten
Finding the Square Root Next, we find the square root of the base-10 number 100:
100 = 10
Converting Back to Base-2 Now, we convert the result, 10, back to base-2:
10 = 8 + 2 = 1 × 2 3 + 0 × 2 2 + 1 × 2 1 + 0 × 2 0
So, 1 0 ten = 101 0 two
Final Answer Therefore, the square root of 110010 0 two is 101 0 two .
Examples
Imagine you're designing digital circuits and need to perform calculations in binary. Finding the square root of a binary number is essential in signal processing or data compression algorithms. For instance, if you have a binary representation of a signal's power, taking the square root helps determine the signal's amplitude. This is crucial for ensuring the signal doesn't exceed the circuit's voltage limits, preventing damage and maintaining signal integrity. In our case, we found that the square root of the binary number 110010 0 two is 101 0 two , which could represent a key parameter in your circuit design.
