The square root division method, also known as the long division method for square roots, is a step-by-step process to find the square root of a number. It's particularly useful for non-perfect squares where estimation might be less precise. Here's a detailed explanation of how to perform this method:
Understanding the Process
The square root division method breaks down into several key steps. It involves pairing digits, finding suitable divisors, and continually refining the quotient until the desired level of accuracy is achieved. This method is an alternative to using a calculator or estimation.
Steps for Long Division Method
Here's how the long division method works, using the example of finding the square root of 529:
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Pair the Digits: Starting from the right, pair the digits of the number under the square root symbol. For 529, we get 5 and 29.
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Identify the Largest Perfect Square:
- Reference: "Take the largest number as the divisor whose square is less than or equal to the number on the extreme left of the number."
- Find the largest perfect square less than or equal to the leftmost pair or single digit (in our case, 5). The perfect square less than 5 is 4, and its square root is 2.
- Write 2 as the first digit of the quotient (the answer). Also, write 2 as the first divisor.
- Subtract the square of 2 (which is 4) from 5. The remainder will be 1.
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Bring Down the Next Pair: Bring down the next pair of digits (29) next to the remainder 1, to make it 129.
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Double the Quotient: Double the current quotient (2), making it 4. This becomes the starting part of our new divisor.
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Find the Next Digit of the Divisor and Quotient: Find a new digit that, when added to the doubled quotient (4), and then this entire new divisor is multiplied by this digit, results in a number less than or equal to 129. In our case, 3 fulfills the condition (43*3 = 129).
- Write 3 as the next digit in the quotient.
- Write 3 next to 4 in divisor (making the divisor 43)
- Multiply the new divisor (43) by 3 and subtract this from 129 and remainder is 0.
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Repeat: If there are more pairs, repeat steps 4 and 5. Since our remainder is zero, we have found the answer.
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Square Root: The square root of 529 is 23.
Example with Decimals
If you need to find the square root of a number with a decimal, add pairs of zeros after the decimal point as needed. The same steps apply, aligning the decimal point in the quotient directly above its position in the original number.
Summary of Steps
| Step | Action | Explanation |
|---|---|---|
| 1 | Pair Digits | Group digits from right to left. |
| 2 | Find First Divisor and Quotient | Identify the largest square less than or equal to the leftmost pair of digits. The square root is first digit of the answer |
| 3 | Bring Down | Carry down the next digit pair. |
| 4 | Double Quotient | Double the current quotient. |
| 5 | Find Next Digit | Determine the next digit that meets the division criteria. |
| 6 | Repeat | Continue the process until the desired accuracy is achieved. |
Key Points:
- The long division method is useful for finding square roots of numbers that do not have obvious integer square roots.
- Understanding the pattern of adding an extra digit to the divisor helps to reduce error in calculation.
- The process requires patience and careful attention to detail to ensure accuracy.
