What is the square root of 163.28? (Show work)

Question

If you can, please explain the basics of square roots.

Answer 1

There is another way to directly calculate square roots which is very similar to long division:

The first thing about this method is that numbers are "brought down" 2 at a time instead of 1 as in long division. Think of pairs of digits starting from the decimal point going each way.

 

√163.28  think of as √1 63. 28

 

The first "group" is the 1. Find the closest square root, which is 1. Write a 1 above the 1 under the square root.

 

  1

√1 63. 28       Now double what ever is on "top" and put to the side and leave a space for an extra digit.

 

      1

2_ √1 63. 28

       1

       0 63         1 squared is one, subtract from 1 which is 0. Bring down the next "group" which is 63

 

Now it will look even more like long division with a twist. Find a number 2_ (twenty something), when multiplied by the last digit will equal or almost equal 63 without going over. Example: 21*1, 22*2, 23*3

 

23*3 is close but gives us 69, too much, so we have to use 22*2 = 44. Subtract 44 from 63, get the result, and bring down the next group. And put a 2 above the 63 because we did 22*2

 

       1  2.
22 √1 63. 28
       1
       0 63

          44

          19 28       Now once again, double what is on top and leave room for an extra digit

 

       1  2.
22 √1 63. 28
       1
       0 63
          44
24_     19 28       Now think about two hundred forty something times somethng to get near 1928

 

                         Let's try 248*8, 248*8 = 1984, too large, so let's use 247*7=1729. Put 7 on top.

 

 

       1  2.   7
22 √1 63. 28
       1
       0 63
          44
247     19 28

          17 29 

254_     1 99 00    Double what is on top, leave an extra digit and keep going!

 

 

       1   2.  7   7  8   0   1
22 √1 63. 28 00 00 00 00
       1
       0 63
          44
247     19 28
          17 29
2547     1 99 00                 2547*7 = 17829

            1 78 29

25548      20 71 00             25548*8 = 204384

               20 43 84

255560         27 16            255561*1 is too large so use 0

                          0

2555600       27 16 00        2555601*1 is too large so use 0 again

                              0

25556001      27 16 00 00   25556001*1= 25556001

                    25 55 60 01

                      1 60 39 99

 

So out to 5 decimal places we have √163.28 ≈ 12.77801

 

 

 

 

 

Answer 2

The square roots of a number x are those numbers that, when squared, will give back x. For example, the square roots of 4 are 2 and -2, because 2²=(-2)²=4. Any positive number will have a positive and a negative square root of the same absolute value.

 

You find the positive square root of 163.28 in steps, each of which will give a more accurate answer. Start with two numbers, one of which has a square smaller than 163.28 and the other has a larger square.

 

You know that 12²=144 and 13²=169. Since

 

12² < 163.28 < 13², the positive square root must lie between 12 and 13. We write

12 < √163.28 < 13

 

Now repeat the process with some number between 12 and 13, say 12.5. Since 12.5²=156.25, we have

 

12.5² < 163.28 < 13², or

12.5 < √163.28 < 13

 

This way you get as close as you like to the exact answer (which may be an irrational number).

 

With 5 significant digits accuracy, you will find

 

12.778 < √163.28 < 12.779