Question

Use the babylonian method to approximate square root of 24 to the nearest hundredth.

Answer 1

We will start with our guess of 12, since 12*2 = 24.
Divide 24 by 12; 24/12=2.  Average this answer with our guess:  (12+2)/2=7.  This is our new guess.
24/7=3.428571429.  Average this with our guess of 7:  (3.428571429+7)/2=5.214285715.  This is our new guess.
24/5.214285715=4.602739726.  Averaging with our guess:  (4.602739726+5.214285715)/2=4.90851272.  New guess!
24/4.90851272=4.889464766.  Averaging with our guess:  (4.889464766+4.90851272)/2=4.898988743.  New guess!  You can see as we go through our guesses are closer and closer to the same number...)
24/4.898988743=4.898970228.  Averaging:  (4.898970228+4.898988743)/2=4.898979486.  At this point our answer is the same every time down to the hundred-thousandth.  Our estimate to the nearest hundredth would be 4.90.