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The repeated subtraction method is one of the simplest way to find the square root of a number.
We know that the square number can be written as the sum of consecutive odd numbers starting from \(1\).
We can obtain the square root of a number by repeatedly subtracting the successive odd numbers (starting from \(1\)) from the given square number till we reach zero.
At one particular step, we get zero. The step where we get zero is the square root of a given number.
If we did not get zero, the given number is not a perfect square.
Let's see an example to understand this concept.
Example:
Obtain the square root of \(49\) by repeated subtraction method.
Solution:
The given number is \(49\).
We need to find the square root of \(49\) by the repeated subtraction method.
Subtract the given number by the odd numbers starting from \(1\) till we get \(0\).
Step 1: \(49 - 1 = 48\)
Step 2: \(48 - 3 = 45\)
Step 3: \(45 - 5 = 40\)
Step 4: \(40 - 7 = 33\)
Step 5: \(33 - 9 = 24\)
Step 6: \(24 - 11 = 13\)
Step 7: \(13 - 13 = 0\)
Here we get zero on the \(7^{\text{th}}\) step. That is \(\sqrt{49} = 7\).
Therefore, the square root of \(49\) is \(7\).
Can we find a square root of \(15876\) by repeated subtraction method?
Yes, we can, but if we use this method for a large number like \(15876\), we can find the square root, but it will take a huge time. So, we use this method only for small numbers.
We will use some other methods, like prime factorization and long division, to find the given number's square root.
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