Question

find the square root of 152.7696​

Answer 1

Answer:

The square root of 152.7 is 12.357184145265. Or,

√152.7 = 12.357184145265

See, below, details on how to calculate this square root using the Babylonian Method

Step-by-step explanation:

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 (152.7) by 2 to get the first guess for the square root .

First guess = 152.7/2 = 76.35.

Step 2: Divide 152.7 by the previous result. d = 152.7/76.35 = 2.

Average this value (d) with that of step 1: (2 + 76.35)/2 = 39.175 (new guess).

Error = new guess - previous value = 76.35 - 39.175 = 37.175.

37.175 > 0.001. As error > accuracy, we repeat this step again.

Step 3: Divide 152.7 by the previous result. d = 152.7/39.175 = 3.8978940651.

Average this value (d) with that of step 2: (3.8978940651 + 39.175)/2 = 21.5364470325 (new guess). Error = new guess - previous value = 39.175 - 21.5364470325 = 17.6385529675.

17.6385529675 > 0.001. As error > accuracy, we repeat this step again.

Step 4: Divide 152.7 by the previous result. d = 152.7/21.5364470325 = 7.090306018.

Average this value (d) with that of step 3: (7.090306018 + 21.5364470325)/2 = 14.3133765253 (new guess).

Error = new guess - previous value = 21.5364470325 - 14.3133765253 = 7.2230705072.

7.2230705072 > 0.001. As error > accuracy, we repeat this step again.

Step 5: Divide 152.7 by the previous result. d = 152.7/14.3133765253 = 10.6683422832.

Average this value (d) with that of step 4: (10.6683422832 + 14.3133765253)/2 = 12.4908594042 (new guess).

Error = new guess - previous value = 14.3133765253 - 12.4908594042 = 1.8225171211.

1.8225171211 > 0.001. As error > accuracy, we repeat this step again.

Step 6: Divide 152.7 by the previous result. d = 152.7/12.4908594042 = 12.2249394584.

Average this value (d) with that of step 5: (12.2249394584 + 12.4908594042)/2 = 12.3578994313 (new guess).

Error = new guess - previous value = 12.4908594042 - 12.3578994313 = 0.1329599729. 0.1329599729 > 0.001. As error > accuracy, we repeat this step again.

Step 7: Divide 152.7 by the previous result. d = 152.7/12.3578994313 = 12.3564689006.

Average this value (d) with that of step 6: (12.3564689006 + 12.3578994313)/2 = 12.3571841659 (new guess).

Error = new guess - previous value = 12.3578994313 - 12.3571841659 = 0.0007152654. 0.0007152654 <= 0.001. As error <= accuracy, we stop the iterations and use 12.3571841659 as the square root.

So, we can say that the square root of 152.7 is 12.357