Square root of 8000 long division method
Answers
Answer : Step 1:
Divide the number (8000) by 2 to get the first guess for the square root .
First guess = 8000/2 = 4000.
Step 2:
Divide 8000 by the previous result. d = 8000/4000 = 2.
Average this value (d) with that of step 1: (2 + 4000)/2 = 2001 (new guess).
Error = new guess - previous value = 4000 - 2001 = 1999.
1999 > 0.001. As error > accuracy, we repeat this step again.
Step 3:
Divide 8000 by the previous result. d = 8000/2001 = 3.9980009995.
Average this value (d) with that of step 2: (3.9980009995 + 2001)/2 = 1002.4990004998 (new guess).
Error = new guess - previous value = 2001 - 1002.4990004998 = 998.5009995002.
998.5009995002 > 0.001. As error > accuracy, we repeat this step again.
Step 4:
Divide 8000 by the previous result. d = 8000/1002.4990004998 = 7.9800578315.
Average this value (d) with that of step 3: (7.9800578315 + 1002.4990004998)/2 = 505.2395291657 (new guess).
Error = new guess - previous value = 1002.4990004998 - 505.2395291657 = 497.2594713341.
497.2594713341 > 0.001. As error > accuracy, we repeat this step again.
Step 5:
Divide 8000 by the previous result. d = 8000/505.2395291657 = 15.8340738168.
Average this value (d) with that of step 4: (15.8340738168 + 505.2395291657)/2 = 260.5368014913 (new guess).
Error = new guess - previous value = 505.2395291657 - 260.5368014913 = 244.7027276744.
244.7027276744 > 0.001. As error > accuracy, we repeat this step again.
Step 6:
Divide 8000 by the previous result. d = 8000/260.5368014913 = 30.7058348541.
Average this value (d) with that of step 5: (30.7058348541 + 260.5368014913)/2 = 145.6213181727 (new guess).
Error = new guess - previous value = 260.5368014913 - 145.6213181727 = 114.9154833186.
114.9154833186 > 0.001. As error > accuracy, we repeat this step again.
Step 7:
Divide 8000 by the previous result. d = 8000/145.6213181727 = 54.9370112864.
Average this value (d) with that of step 6: (54.9370112864 + 145.6213181727)/2 = 100.2791647296 (new guess).
Error = new guess - previous value = 145.6213181727 - 100.2791647296 = 45.3421534431.
45.3421534431 > 0.001. As error > accuracy, we repeat this step again.
Step 8:
Divide 8000 by the previous result. d = 8000/100.2791647296 = 79.7772899442.
Average this value (d) with that of step 7: (79.7772899442 + 100.2791647296)/2 = 90.0282273369 (new guess).
Error = new guess - previous value = 100.2791647296 - 90.0282273369 = 10.2509373927.
10.2509373927 > 0.001. As error > accuracy, we repeat this step again.
Step 9:
Divide 8000 by the previous result. d = 8000/90.0282273369 = 88.8610187787.
Average this value (d) with that of step 8: (88.8610187787 + 90.0282273369)/2 = 89.4446230578 (new guess).
Error = new guess - previous value = 90.0282273369 - 89.4446230578 = 0.5836042791.
0.5836042791 > 0.001. As error > accuracy, we repeat this step again.
Step 10:
Divide 8000 by the previous result. d = 8000/89.4446230578 = 89.4408151827.
Average this value (d) with that of step 9: (89.4408151827 + 89.4446230578)/2 = 89.4427191203 (new guess).
Error = new guess - previous value = 89.4446230578 - 89.4427191203 = 0.0019039375.
0.0019039375 > 0.001. As error > accuracy, we repeat this step again.
Step 11:
Divide 8000 by the previous result. d = 8000/89.4427191203 = 89.4427190797.
Average this value (d) with that of step 10: (89.4427190797 + 89.4427191203)/2 = 89.4427191 (new guess).
Error = new guess - previous value = 89.4427191203 - 89.4427191 = 2.03e-8.
2.03e-8 <= 0.001. As error <= accuracy, we stop the iterations and use 89.4427191 as the square root.
So, we can say that the square root of 8000 is 89.4427191 with an error smaller than 0.001