Question

find the square root of 18.49 by prime factorization
show by step​

Answer 1

Step-by-step explanation:

Step 1:

Divide the number (18.49) by 2 to get the first guess for the square root .

First guess = 18.49/2 = 9.245.

Step 2:

Divide 18.49 by the previous result. d = 18.49/9.245 = 2.

Average this value (d) with that of step 1: (2 + 9.245)/2 = 5.6225 (new guess).

Error = new guess - previous value = 9.245 - 5.6225 = 3.6225.

3.6225 > 0.001. As error > accuracy, we repeat this step again.

Step 3:

Divide 18.49 by the previous result. d = 18.49/5.6225 = 3.288572699.

Average this value (d) with that of step 2: (3.288572699 + 5.6225)/2 = 4.4555363495 (new guess).

Error = new guess - previous value = 5.6225 - 4.4555363495 = 1.1669636505.

1.1669636505 > 0.001. As error > accuracy, we repeat this step again.

Step 4:

Divide 18.49 by the previous result. d = 18.49/4.4555363495 = 4.1498932002.

Average this value (d) with that of step 3: (4.1498932002 + 4.4555363495)/2 = 4.3027147749 (new guess).

Error = new guess - previous value = 4.4555363495 - 4.3027147749 = 0.1528215746.

0.1528215746 > 0.001. As error > accuracy, we repeat this step again.

Step 5:

Divide 18.49 by the previous result. d = 18.49/4.3027147749 = 4.297286938.

Average this value (d) with that of step 4: (4.297286938 + 4.3027147749)/2 = 4.3000008565 (new guess).

Error = new guess - previous value = 4.3027147749 - 4.3000008565 = 0.0027139184.

0.0027139184 > 0.001. As error > accuracy, we repeat this step again.

Step 6:

Divide 18.49 by the previous result. d = 18.49/4.3000008565 = 4.2999991435.

Average this value (d) with that of step 5: (4.2999991435 + 4.3000008565)/2 = 4.3 (new guess).

Error = new guess - previous value = 4.3000008565 - 4.3 = 8.565e-7.

8.565e-7 <= 0.001. As error <= accuracy, we stop the iterations and use 4.3 as the square root.

So, we can say that the square root of 18.49 is 4.3 with an error smaller than 0.001 (in fact the error is 8.565e-7). this means that the first 6 decimal places are correct. Just to compare, the returned value by using the javascript function 'Math.sqrt(18.49)' is 4.3.

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