Question

how to find root 5000

Answer 1

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 (5000) by 2 to get the first guess for the square root .
 First guess = 5000/2 = 2500.

Step 2:
 Divide 5000 by the previous result. d = 5000/2500 = 2.
 Average this value (d) with that of step 1: (2 + 2500)/2 = 1251 (new guess).
 Error = new guess - previous value = 2500 - 1251 = 1249.
 1249 > 0.001. As error > accuracy, we repeat this step again.

Step 3:
 Divide 5000 by the previous result. d = 5000/1251 = 3.996802558.
 Average this value (d) with that of step 2: (3.996802558 + 1251)/2 = 627.498401279 (new guess).
 Error = new guess - previous value = 1251 - 627.498401279 = 623.501598721.
 623.501598721 > 0.001. As error > accuracy, we repeat this step again.

Step 4:
 Divide 5000 by the previous result. d = 5000/627.498401279 = 7.968147791.
 Average this value (d) with that of step 3: (7.968147791 + 627.498401279)/2 = 317.733274535 (new guess).
 Error = new guess - previous value = 627.498401279 - 317.733274535 = 309.765126744.
 309.765126744 > 0.001. As error > accuracy, we repeat this step again.

Step 5:
 Divide 5000 by the previous result. d = 5000/317.733274535 = 15.7364695508.
 Average this value (d) with that of step 4: (15.7364695508 + 317.733274535)/2 = 166.7348720429 (new guess).
 Error = new guess - previous value = 317.733274535 - 166.7348720429 = 150.9984024921.
 150.9984024921 > 0.001. As error > accuracy, we repeat this step again.

Step 6:
 Divide 5000 by the previous result. d = 5000/166.7348720429 = 29.9877280544.
 Average this value (d) with that of step 5: (29.9877280544 + 166.7348720429)/2 = 98.3613000487 (new guess).
 Error = new guess - previous value = 166.7348720429 - 98.3613000487 = 68.3735719942.
 68.3735719942 > 0.001. As error > accuracy, we repeat this step again.

Step 7:
 Divide 5000 by the previous result. d = 5000/98.3613000487 = 50.833000352.
 Average this value (d) with that of step 6: (50.833000352 + 98.3613000487)/2 = 74.5971502004 (new guess).
 Error = new guess - previous value = 98.3613000487 - 74.5971502004 = 23.7641498483.
 23.7641498483 > 0.001. As error > accuracy, we repeat this step again.

Step 8:
 Divide 5000 by the previous result. d = 5000/74.5971502004 = 67.0266891774.
 Average this value (d) with that of step 7: (67.0266891774 + 74.5971502004)/2 = 70.8119196889 (new guess).
 Error = new guess - previous value = 74.5971502004 - 70.8119196889 = 3.7852305115.
 3.7852305115 > 0.001. As error > accuracy, we repeat this step again.

Step 9:
 Divide 5000 by the previous result. d = 5000/70.8119196889 = 70.609581296.
 Average this value (d) with that of step 8: (70.609581296 + 70.8119196889)/2 = 70.7107504925 (new guess).
 Error = new guess - previous value = 70.8119196889 - 70.7107504925 = 0.1011691964.
 0.1011691964 > 0.001. As error > accuracy, we repeat this step again.

Step 10:
 Divide 5000 by the previous result. d = 5000/70.7107504925 = 70.7106057449.
 Average this value (d) with that of step 9: (70.7106057449 + 70.7107504925)/2 = 70.7106781187 (new guess).
 Error = new guess - previous value = 70.7107504925 - 70.7106781187 = 0.0000723738.
 0.0000723738 <= 0.001. As error <= accuracy, we stop the iterations and use 70.7106781187 as the square root.

So, we can say that the square root of 5000 is 70.7106 with an error smaller than 0.001 (in fact the error is 0.0000723738). this means that the first 4 decimal places are correct. Just to compare, the returned value by using the javascript function 'Math.sqrt(5000)' is 70.71067811865476.

Answer 2

Under root of 5000 is
50 under root 2
For this we making the factors inside the root then the common factored can be taken out