find square root of 256000 by long division method
step by step explanation
Answers
Step 1:
Divide the number (256000) by 2 to get the first guess for the square root .
First guess = 256000/2 = 128000.
Step 2:
Divide 256000 by the previous result. d = 256000/128000 = 2.
Average this value (d) with that of step 1: (2 + 128000)/2 = 64001 (new guess).
Error = new guess - previous value = 128000 - 64001 = 63999.
63999 > 0.001. As error > accuracy, we repeat this step again.
Step 3:
Divide 256000 by the previous result. d = 256000/64001 = 3.999937501.
Average this value (d) with that of step 2: (3.999937501 + 64001)/2 = 32002.4999687505 (new guess).
Error = new guess - previous value = 64001 - 32002.4999687505 = 31998.5000312495.
31998.5000312495 > 0.001. As error > accuracy, we repeat this step again.
Step 4:
Divide 256000 by the previous result. d = 256000/32002.4999687505 = 7.9993750566.
Average this value (d) with that of step 3: (7.9993750566 + 32002.4999687505)/2 = 16005.2496719036 (new guess).
Error = new guess - previous value = 32002.4999687505 - 16005.2496719036 = 15997.2502968469.
15997.2502968469 > 0.001. As error > accuracy, we repeat this step again.
Step 5:
Divide 256000 by the previous result. d = 256000/16005.2496719036 = 15.99475205.
Average this value (d) with that of step 4: (15.99475205 + 16005.2496719036)/2 = 8010.6222119768 (new guess).
Error = new guess - previous value = 16005.2496719036 - 8010.6222119768 = 7994.6274599268.
7994.6274599268 > 0.001. As error > accuracy, we repeat this step again.
Step 6:
Divide 256000 by the previous result. d = 256000/8010.6222119768 = 31.957567493.
Average this value (d) with that of step 5: (31.957567493 + 8010.6222119768)/2 = 4021.2898897349 (new guess).
Error = new guess - previous value = 8010.6222119768 - 4021.2898897349 = 3989.3323222419.
3989.3323222419 > 0.001. As error > accuracy, we repeat this step again.
Step 7:
Divide 256000 by the previous result. d = 256000/4021.2898897349 = 63.6611652031.
Average this value (d) with that of step 6: (63.6611652031 + 4021.2898897349)/2 = 2042.475527469 (new guess).
Error = new guess - previous value = 4021.2898897349 - 2042.475527469 = 1978.8143622659.
1978.8143622659 > 0.001. As error > accuracy, we repeat this step again.
Step 8:
Divide 256000 by the previous result. d = 256000/2042.475527469 = 125.3380990651.
Average this value (d) with that of step 7: (125.3380990651 + 2042.475527469)/2 = 1083.9068132671 (new guess).
Error = new guess - previous value = 2042.475527469 - 1083.9068132671 = 958.5687142019.
958.5687142019 > 0.001. As error > accuracy, we repeat this step again.
Step 9:
Divide 256000 by the previous result. d = 256000/1083.9068132671 = 236.182665213.
Average this value (d) with that of step 8: (236.182665213 + 1083.9068132671)/2 = 660.0447392401 (new guess).
Error = new guess - previous value = 1083.9068132671 - 660.0447392401 = 423.862074027.
423.862074027 > 0.001. As error > accuracy, we repeat this step again.
Step 10:
Divide 256000 by the previous result. d = 256000/660.0447392401 = 387.8524966273.
Average this value (d) with that of step 9: (387.8524966273 + 660.0447392401)/2 = 523.9486179337 (new guess).
Error = new guess - previous value = 660.0447392401 - 523.9486179337 = 136.0961213064.
136.0961213064 > 0.001. As error > accuracy, we repeat this step again.
Step 11:
Divide 256000 by the previous result. d = 256000/523.9486179337 = 488.5975289134.
Average this value (d) with that of step 10: (488.5975289134 + 523.9486179337)/2 = 506.2730734236 (new guess).
Error = new guess - previous value = 523.9486179337 - 506.2730734236 = 17.6755445101.
17.6755445101 > 0.001. As error > accuracy, we repeat this step again.
Step 12:
Divide 256000 by the previous result. d = 256000/506.2730734236 = 505.6559659964.
Average this value (d) with that of step 11: (505.6559659964 + 506.2730734236)/2 = 505.96451971 (new guess).
Error = new guess - previous value = 506.2730734236 - 505.96451971 = 0.3085537136.
0.3085537136 > 0.001. As error > accuracy, we repeat this step again.
Step 13:
Divide 256000 by the previous result. d = 256000/505.96451971 = 505.9643315439.
Average this value (d) with that of step 12: (505.9643315439 + 505.96451971)/2 = 505.964425627 (new guess).
Error = new guess - previous value = 505.96451971 - 505.964425627 = 0.000094083.
0.000094083 <= 0.001. As error <= accuracy, we stop the iterations and use 505.964425627 as the square root.
So, we can say that the square root of 256000 is 505.9644 with an error smaller than 0.001 (in fact the error is 0.000094083). this means that the first 4 decimal places are correct. Just to compare, the returned value by using the javascript function 'Math.sqrt(256000)' is 505.9644256269407.