Using nonperfect square method,find the square root of 418 correct to two decimal places, showing workings.
Newton's method of square root estimation often leads directly to the minimal solution.
1--Make an estimate of the square root of N = n.
2--Then, calculate n1 = [N/n + n]/2
3--For further accuracy, calculate n2 = [N/n1 + n1]/2
Example: Estimate the square root of 73.
n1 = [73/8 + 8]/2 = 8.5625
n2 = [73/8.5625 + 8.5625]/2 = 8.54402
The actural square root of 73 is 8.544003745.
The method gives a reasonably close estimate in two calculations.
squar root of 2.5
To find the square root of a number using the non-perfect square method, we will use a process called long division. Here's how you can find the square root of 418:
1. Start by separating the number into pairs of digits from right to left.
4 18
2. Find the largest perfect square, less than or equal to the leftmost pair (in this case, 4). The largest perfect square less than 4 is 2.
√4 = 2
3. Write the 2 above the leftmost pair, and subtract the square of 2 from the leftmost pair.
2
- 4
------
18
4. Bring down the next pair of digits (18) and double the number above the line (the 2).
2 | 18
5. Append a digit (we'll call it 'x') to the current quotient, making it 2x. Then, find the largest digit 'x' that satisfies (20x) * x ≤ 180. In this case, 'x' would be 8, as (208 * 8) is less than or equal to 180.
2 | 18
8x
6. Append the 'x' digit to the right of the current quotient, making it 28. Now, divide 208 by 28, and find the largest digit 'y' that satisfies (28y) * y ≤ 208. In this case, 'y' would be 2, as (282 * 2) is less than or equal to 208.
28 | 180
2 (28 * 2 = 56)
7. Append the 'y' digit to the right of the current quotient, making it 282. Now, divide 208 by 282, and find the largest digit 'z' that satisfies (282z) * z ≤ 208. In this case, 'z' would be 3, as (2823 * 3) is less than or equal to 208.
2823 | 180
2
5 (2823 * 3 = 8469)
8. Repeat this process until you have a sufficient number of decimal places after the square root. The more decimal places you calculate, the closer the approximation will be to the accurate square root.
In this example, the square root of 418, correct to two decimal places, using the non-perfect square method, is approximately 20.49.