square root of 4097 using division method
Answers
Answer:
64.0078120232
Step-by-step explanation:
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Answer:
The square root of 4097 is 64.007812023221
Step-by-step explanation:
Step 1:
Divide the number (4097) by 2 to get the first guess for the square root .
First guess = 4097/2 = 2048.5.
Step 2:
Divide 4097 by the previous result. d = 4097/2048.5 = 2.
Average this value (d) with that of step 1: (2 + 2048.5)/2 = 1025.25 (new guess).
Error = new guess - previous value = 2048.5 - 1025.25 = 1023.25.
1023.25 > 0.001. As error > accuracy, we repeat this step again.
Step 3:
Divide 4097 by the previous result. d = 4097/1025.25 = 3.9960985126.
Average this value (d) with that of step 2: (3.9960985126 + 1025.25)/2 = 514.6230492563 (new guess).
Error = new guess - previous value = 1025.25 - 514.6230492563 = 510.6269507437.
510.6269507437 > 0.001. As error > accuracy, we repeat this step again.
Step 4:
Divide 4097 by the previous result. d = 4097/514.6230492563 = 7.9611669278.
Average this value (d) with that of step 3: (7.9611669278 + 514.6230492563)/2 = 261.2921080921 (new guess).
Error = new guess - previous value = 514.6230492563 - 261.2921080921 = 253.3309411642.
253.3309411642 > 0.001. As error > accuracy, we repeat this step again.
Step 5:
Divide 4097 by the previous result. d = 4097/261.2921080921 = 15.6797693965.
Average this value (d) with that of step 4: (15.6797693965 + 261.2921080921)/2 = 138.4859387443 (new guess).
Error = new guess - previous value = 261.2921080921 - 138.4859387443 = 122.8061693478.
122.8061693478 > 0.001. As error > accuracy, we repeat this step again.
Step 6:
Divide 4097 by the previous result. d = 4097/138.4859387443 = 29.5842309851.
Average this value (d) with that of step 5: (29.5842309851 + 138.4859387443)/2 = 84.0350848647 (new guess).
Error = new guess - previous value = 138.4859387443 - 84.0350848647 = 54.4508538796.
54.4508538796 > 0.001. As error > accuracy, we repeat this step again.
Step 7:
Divide 4097 by the previous result. d = 4097/84.0350848647 = 48.7534463325.
Average this value (d) with that of step 6: (48.7534463325 + 84.0350848647)/2 = 66.3942655986 (new guess).
Error = new guess - previous value = 84.0350848647 - 66.3942655986 = 17.6408192661.
17.6408192661 > 0.001. As error > accuracy, we repeat this step again.
Step 8:
Divide 4097 by the previous result. d = 4097/66.3942655986 = 61.7071363477.
Average this value (d) with that of step 7: (61.7071363477 + 66.3942655986)/2 = 64.0507009732 (new guess).
Error = new guess - previous value = 66.3942655986 - 64.0507009732 = 2.3435646254.
2.3435646254 > 0.001. As error > accuracy, we repeat this step again.
Step 9:
Divide 4097 by the previous result. d = 4097/64.0507009732 = 63.9649517921.
Average this value (d) with that of step 8: (63.9649517921 + 64.0507009732)/2 = 64.0078263827 (new guess).
Error = new guess - previous value = 64.0507009732 - 64.0078263827 = 0.0428745905.
0.0428745905 > 0.001. As error > accuracy, we repeat this step again.
Step 10:
Divide 4097 by the previous result. d = 4097/64.0078263827 = 64.0077976637.
Average this value (d) with that of step 9: (64.0077976637 + 64.0078263827)/2 = 64.0078120232 (new guess).
Error = new guess - previous value = 64.0078263827 - 64.0078120232 = 0.0000143595.
0.0000143595 <= 0.001. As error <= accuracy, we stop the iterations and use 64.0078120232 as the square root.
So, we can say that the square root of 4097 is 64.0078 with an error smaller than 0.001 (in fact the error is 0.0000143595). this means that the first 4 decimal places are correct. Just to compare, the returned value by using the javascript function 'Math.sqrt(4097)' is 64.00781202322104.
Note: There are other ways to calculate square roots. This is only one of them.
What is square root?
Definition of square root
A square root of a number 'a' is a number x such that x2 = a, in other words, a number x whose square is a. For example, 64 is the square root of 4096 because 642 = 64•64 = 4096, -64 is square root of 4096 because (-64)2 = (-64)•(-64) = 4096.