square root of 6529 by long division method
Answers
Answer:
square root of 6529 is 80.80222769206304
In this case we are going to use the 'Babylonian Method' to get the square root of any positive number.
We must set an error for the final result. Say, smaller than 0.001. In other words we will try to find the square root value with at least 2 correct decimal places.
Step 1:
Divide the number (6529) by 2 to get the first guess for the square root .
First guess = 6529/2 = 3264.5.
Step 2:
Divide 6529 by the previous result. d = 6529/3264.5 = 2.
Average this value (d) with that of step 1: (2 + 3264.5)/2 = 1633.25 (new guess).
Error = new guess - previous value = 3264.5 - 1633.25 = 1631.25.
1631.25 > 0.001. As error > accuracy, we repeat this step again.
Step 3:
Divide 6529 by the previous result. d = 6529/1633.25 = 3.9975508955.
Average this value (d) with that of step 2: (3.9975508955 + 1633.25)/2 = 818.6237754477 (new guess).
Error = new guess - previous value = 1633.25 - 818.6237754477 = 814.6262245523.
814.6262245523 > 0.001. As error > accuracy, we repeat this step again.
Step 4:
Divide 6529 by the previous result. d = 6529/818.6237754477 = 7.9755807195.
Average this value (d) with that of step 3: (7.9755807195 + 818.6237754477)/2 = 413.2996780836 (new guess).
Error = new guess - previous value = 818.6237754477 - 413.2996780836 = 405.3240973641.
405.3240973641 > 0.001. As error > accuracy, we repeat this step again.
Step 5:
Divide 6529 by the previous result. d = 6529/413.2996780836 = 15.7972540174.
Average this value (d) with that of step 4: (15.7972540174 + 413.2996780836)/2 = 214.5484660505 (new guess).
Error = new guess - previous value = 413.2996780836 - 214.5484660505 = 198.7512120331.
198.7512120331 > 0.001. As error > accuracy, we repeat this step again.
Step 6:
Divide 6529 by the previous result. d = 6529/214.5484660505 = 30.431352506.
Average this value (d) with that of step 5: (30.431352506 + 214.5484660505)/2 = 122.4899092783 (new guess).
Error = new guess - previous value = 214.5484660505 - 122.4899092783 = 92.0585567722.
92.0585567722 > 0.001. As error > accuracy, we repeat this step again.
Step 7:
Divide 6529 by the previous result. d = 6529/122.4899092783 = 53.3023498708.
Average this value (d) with that of step 6: (53.3023498708 + 122.4899092783)/2 = 87.8961295746 (new guess).
Error = new guess - previous value = 122.4899092783 - 87.8961295746 = 34.5937797037.
34.5937797037 > 0.001. As error > accuracy, we repeat this step again.
Step 8:
Divide 6529 by the previous result. d = 6529/87.8961295746 = 74.2808589138.
Average this value (d) with that of step 7: (74.2808589138 + 87.8961295746)/2 = 81.0884942442 (new guess).
Error = new guess - previous value = 87.8961295746 - 81.0884942442 = 6.8076353304.
6.8076353304 > 0.001. As error > accuracy, we repeat this step again.
Step 9:
Divide 6529 by the previous result. d = 6529/81.0884942442 = 80.5169717462.
Average this value (d) with that of step 8: (80.5169717462 + 81.0884942442)/2 = 80.8027329952 (new guess).
Error = new guess - previous value = 81.0884942442 - 80.8027329952 = 0.285761249.
0.285761249 > 0.001. As error > accuracy, we repeat this step again.
Step 10:
Divide 6529 by the previous result. d = 6529/80.8027329952 = 80.8017223921.
Average this value (d) with that of step 9: (80.8017223921 + 80.8027329952)/2 = 80.8022276937 (new guess).
Error = new guess - previous value = 80.8027329952 - 80.8022276937 = 0.0005053015.
0.0005053015 <= 0.001. As error <= accuracy, we stop the iterations and use 80.8022276937 as the square root.
So, we can say that the square root of 6529 is 80.802 with an error smaller than 0.001 (in fact the error is 0.0005053015). this means that the first 3 decimal places are correct. Just to compare, the returned value by using the javascript function 'Math.sqrt(6529)' is 80.80222769206304.