Question

Square root of 81.25 by long division method with full steps

Answer 1

Step-by-step explanation:

Step 1:

Divide the number (81.25) by 2 to get the first guess for the square root .

First guess = 81.25/2 = 40.625.

Step 2:

Divide 81.25 by the previous result. d = 81.25/40.625 = 2.

Average this value (d) with that of step 1: (2 + 40.625)/2 = 21.3125 (new guess).

Error = new guess - previous value = 40.625 - 21.3125 = 19.3125.

19.3125 > 0.001. As error > accuracy, we repeat this step again.

Step 3:

Divide 81.25 by the previous result. d = 81.25/21.3125 = 3.8123167155.

Average this value (d) with that of step 2: (3.8123167155 + 21.3125)/2 = 12.5624083578 (new guess).

Error = new guess - previous value = 21.3125 - 12.5624083578 = 8.7500916422.

8.7500916422 > 0.001. As error > accuracy, we repeat this step again.

Step 4:

Divide 81.25 by the previous result. d = 81.25/12.5624083578 = 6.4677088728.

Average this value (d) with that of step 3: (6.4677088728 + 12.5624083578)/2 = 9.5150586153 (new guess).

Error = new guess - previous value = 12.5624083578 - 9.5150586153 = 3.0473497425.

3.0473497425 > 0.001. As error > accuracy, we repeat this step again.

Step 5:

Divide 81.25 by the previous result. d = 81.25/9.5150586153 = 8.5390961091.

Average this value (d) with that of step 4: (8.5390961091 + 9.5150586153)/2 = 9.0270773622 (new guess).

Error = new guess - previous value = 9.5150586153 - 9.0270773622 = 0.4879812531.

0.4879812531 > 0.001. As error > accuracy, we repeat this step again.

Step 6:

Divide 81.25 by the previous result. d = 81.25/9.0270773622 = 9.0006983146.

Average this value (d) with that of step 5: (9.0006983146 + 9.0270773622)/2 = 9.0138878384 (new guess).

Error = new guess - previous value = 9.0270773622 - 9.0138878384 = 0.0131895238.

0.0131895238 > 0.001. As error > accuracy, we repeat this step again.

Step 7:

Divide 81.25 by the previous result. d = 81.25/9.0138878384 = 9.0138685389.

Average this value (d) with that of step 6: (9.0138685389 + 9.0138878384)/2 = 9.0138781887 (new guess).

Error = new guess - previous value = 9.0138878384 - 9.0138781887 = 0.0000096497.

0.0000096497 <= 0.001. As error <= accuracy, we stop the iterations and use 9.0138781887 as the square root.

So, we can say that the square root of 81.25 is 9.01387 with an error smaller than 0.001 (in fact the error is 0.0000096497). this means that the first 5 decimal places are correct. Just to compare, the returned value by using the javascript function 'Math.sqrt(81.25)' is 9.013878188659973.

Note: There are other ways to calculate square roots. This is only one of them.