The division method for finding the square root involves a series of steps. Here’s how to apply it to the numbers 0.3249 and 10.24.
Finding the square root of 0.3249
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Pair the numbers: Starting from the decimal point, pair the digits to the left and right. For 0.3249, we have pairs: 0 | 32 | 49.
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Find the largest square: Find the largest integer whose square is less than or equal to 0, which is 0. So, we write down 0 and subtract \(0^2\) from 0, which gives us 0. Bring down the next pair (32), making it 32.
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Double the quotient: Double the current result (0), which is 0, and determine a digit \(x\) such that \(0x \cdot x\) is less than or equal to 32. The digit to use here is 5 because \(05 \cdot 5 = 25\). Write down 5 in the quotient.
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Subtract: Subtract \(25\) from \(32\), resulting in \(32 - 25 = 7\). Bring down the next pair (49), giving 749.
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Double the quotient again: Now, double the current quotient (05) to get 10. We need to find a digit \(y\) such that \(10y \cdot y\) is less than or equal to 749. Testing \(y = 7\), we have \(107 \cdot 7 = 749\).
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Subtract: Subtract \(749\) from \(749\), resulting in \(0\), and thus we’ve completed the square root calculation for 0.3249.
So, the square root of 0.3249 is 0.57.
Finding the square root of 10.24
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Pair the numbers: For 10.24, we pair as follows: 10 | 24.
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Find the largest square: The largest integer whose square is less than or equal to 10 is 3, as \(3^2 = 9\). Write down 3 in the quotient.
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Subtract: Subtract \(9\) from \(10\) to get 1. Bring down the next pair (24) to get 124.
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Double the quotient: Double the current result (3) to get 6. Now, find a digit \(z\) such that \(60z \cdot z\) is less than or equal to 124. Testing \(z = 2\), we have \(62 \cdot 2 = 124\).
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Subtract: Subtract \(124\) from \(124\), resulting in \(0\), completing the square root calculation for 10.24.
So, the square root of 10.24 is 3.2.
Final Results:
- The square root of 0.3249 is 0.57.
- The square root of 10.24 is 3.2.