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

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2 Answers

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
 
 
 
 
 
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