 asked

The square root of 0.007 is 0.08366600265340755. See below the step-by-step solution:

In this case we are going to use the Babylonian Method to get the square root of any positive number.

Step 1:

Divide the number (0.007) by 2 to get the first guess for the square root .
First guess = 0.007/2 = 0.0035.


Step 2:

Divide 0.007 by the previous result. d = 0.007/0.0035 = 2.
Average this value (d) with that of step1: (2 + 0.0035)/2 = 1.00175 (New guess).
Error = current value - previous value = 0.0035 - 1.00175 = 0.99825.
0.99825 > 0.0001. So, error > precision. Let's repeat this step again.


Step 3:

Divide 0.007 by the previous result. d = 0.007/1.00175 = 0.006988.
Average this value (d) with that of step2: (0.006988 + 1.00175)/2 = 0.504369 (New guess).
Error = current value - previous value = 1.00175 - 0.504369 = 0.49738099999999996.
0.49738099999999996 > 0.0001. So, error > precision. Let's repeat this step again.


Step 4:

Divide 0.007 by the previous result. d = 0.007/0.504369 = 0.013879.
Average this value (d) with that of step3: (0.013879 + 0.504369)/2 = 0.259124 (New guess).
Error = current value - previous value = 0.504369 - 0.259124 = 0.24524499999999994.
0.24524499999999994 > 0.0001. So, error > precision. Let's repeat this step again.


Step 5:

Divide 0.007 by the previous result. d = 0.007/0.259124 = 0.027014.
Average this value (d) with that of step4: (0.027014 + 0.259124)/2 = 0.143069 (New guess).
Error = current value - previous value = 0.259124 - 0.143069 = 0.11605500000000002.
0.11605500000000002 > 0.0001. So, error > precision. Let's repeat this step again.


Step 6:

Divide 0.007 by the previous result. d = 0.007/0.143069 = 0.048927.
Average this value (d) with that of step5: (0.048927 + 0.143069)/2 = 0.095998 (New guess).
Error = current value - previous value = 0.143069 - 0.095998 = 0.047071.
0.047071 > 0.0001. So, error > precision. Let's repeat this step again.


Step 7:

Divide 0.007 by the previous result. d = 0.007/0.095998 = 0.072918.
Average this value (d) with that of step6: (0.072918 + 0.095998)/2 = 0.084458 (New guess).
Error = current value - previous value = 0.095998 - 0.084458 = 0.011539999999999995.
0.011539999999999995 > 0.0001. So, error > precision. Let's repeat this step again.


Step 8:

Divide 0.007 by the previous result. d = 0.007/0.084458 = 0.082881.
Average this value (d) with that of step7: (0.082881 + 0.084458)/2 = 0.08367 (New guess).
Error = current value - previous value = 0.084458 - 0.08367 = 0.0007880000000000109.
0.0007880000000000109 > 0.0001. So, error > precision. Let's repeat this step again.


Step 9:

Divide 0.007 by the previous result. d = 0.007/0.08367 = 0.083662.
Average this value (d) with that of step8: (0.083662 + 0.08367)/2 = 0.083666 (New guess).
Error = current value - previous value = 0.08367 - 0.083666 = 0.0000039999999999901226.
0.0000039999999999901226 <= 0.0001. So, error <= precision. We can now stop and use 0.083666 as the square root.


Finally, the square root of 0.007 is 0.083666 with an error smaller than 0.0001.