Question

find the square root of 40400

Answer 1

√40400
=20√101
hope this helps you

Answer 2

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 (40400) by 2 to get the first guess for the square root .
 First guess = 40400/2 = 20200.

Step 2:
 Divide 40400 by the previous result. d = 40400/20200 = 2.
 Average this value (d) with that of step 1: (2 + 20200)/2 = 10101 (new guess).
 Error = new guess - previous value = 20200 - 10101 = 10099.
 10099 > 0.001. As error > accuracy, we repeat this step again.

Step 3:
 Divide 40400 by the previous result. d = 40400/10101 = 3.9996039996.
 Average this value (d) with that of step 2: (3.9996039996 + 10101)/2 = 5052.4998019998 (new guess).
 Error = new guess - previous value = 10101 - 5052.4998019998 = 5048.5001980002.
 5048.5001980002 > 0.001. As error > accuracy, we repeat this step again.

Step 4:
 Divide 40400 by the previous result. d = 40400/5052.4998019998 = 7.9960418769.
 Average this value (d) with that of step 3: (7.9960418769 + 5052.4998019998)/2 = 2530.2479219384 (new guess).
 Error = new guess - previous value = 5052.4998019998 - 2530.2479219384 = 2522.2518800614.
 2522.2518800614 > 0.001. As error > accuracy, we repeat this step again.

Step 5:
 Divide 40400 by the previous result. d = 40400/2530.2479219384 = 15.9668148128.
 Average this value (d) with that of step 4: (15.9668148128 + 2530.2479219384)/2 = 1273.1073683756 (new guess).
 Error = new guess - previous value = 2530.2479219384 - 1273.1073683756 = 1257.1405535628.
 1257.1405535628 > 0.001. As error > accuracy, we repeat this step again.

Step 6:
 Divide 40400 by the previous result. d = 40400/1273.1073683756 = 31.7333800774.
 Average this value (d) with that of step 5: (31.7333800774 + 1273.1073683756)/2 = 652.4203742265 (new guess).
 Error = new guess - previous value = 1273.1073683756 - 652.4203742265 = 620.6869941491.
 620.6869941491 > 0.001. As error > accuracy, we repeat this step again.

Step 7:
 Divide 40400 by the previous result. d = 40400/652.4203742265 = 61.9232654221.
 Average this value (d) with that of step 6: (61.9232654221 + 652.4203742265)/2 = 357.1718198243 (new guess).
 Error = new guess - previous value = 652.4203742265 - 357.1718198243 = 295.2485544022.
 295.2485544022 > 0.001. As error > accuracy, we repeat this step again.

Step 8:
 Divide 40400 by the previous result. d = 40400/357.1718198243 = 113.11082722.
 Average this value (d) with that of step 7: (113.11082722 + 357.1718198243)/2 = 235.1413235222 (new guess).
 Error = new guess - previous value = 357.1718198243 - 235.1413235222 = 122.0304963021.
 122.0304963021 > 0.001. As error > accuracy, we repeat this step again.

Step 9:
 Divide 40400 by the previous result. d = 40400/235.1413235222 = 171.8115701436.
 Average this value (d) with that of step 8: (171.8115701436 + 235.1413235222)/2 = 203.4764468329 (new guess).
 Error = new guess - previous value = 235.1413235222 - 203.4764468329 = 31.6648766893.
 31.6648766893 > 0.001. As error > accuracy, we repeat this step again.

Step 10:
 Divide 40400 by the previous result. d = 40400/203.4764468329 = 198.5487786367.
 Average this value (d) with that of step 9: (198.5487786367 + 203.4764468329)/2 = 201.0126127348 (new guess).
 Error = new guess - previous value = 203.4764468329 - 201.0126127348 = 2.4638340981.
 2.4638340981 > 0.001. As error > accuracy, we repeat this step again.

Step 11:
 Divide 40400 by the previous result. d = 40400/201.0126127348 = 200.9824132444.
 Average this value (d) with that of step 10: (200.9824132444 + 201.0126127348)/2 = 200.9975129896 (new guess).
 Error = new guess - previous value = 201.0126127348 - 200.9975129896 = 0.0150997452.
 0.0150997452 > 0.001. As error > accuracy, we repeat this step again.

Step 12:
 Divide 40400 by the previous result. d = 40400/200.9975129896 = 200.9975118552.
 Average this value (d) with that of step 11: (200.9975118552 + 200.9975129896)/2 = 200.9975124224 (new guess).
 Error = new guess - previous value = 200.9975129896 - 200.9975124224 = 5.672e-7.
 5.672e-7 <= 0.001. As error <= accuracy, we stop the iterations and use 200.9975124224 as the square root.

So, we can say that the square root of 40400 is 200.997512 with an error smaller than 0.001 (in fact the error is 5.672e-7). this means that the first 6 decimal places are correct. Just to compare, the returned value by using the javascript function 'Math.sqrt(40400)' is 200.9975124224178

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Answer is 200.9975124224178