<PRE><B>Question 75646</B>
&lt;pre&gt;&lt;font size = 5&gt;&lt;b&gt;
Back in the good old days, they taught us a
&#39;long hand&#39; method for solving Square, Cube,
and Nth roots of any number. For instance 
square root of 139, or 6th root of 13. I have 
since forgotten (after 35years) how to do that 
and want to teach my son, rather than the 
current approach of memorizing your perfect 
squares, cubes, etc. Those formulas exist and 
are embedded in modern scientific calculators, 
but I am unable to find them in any textbook, 
online, etc. Please remind me how to do this. 

OK, but it&#39;s hard! These days a calculator
is the wisest choice for finding square roots.
But here is how it was done in the &quot;good old
days&quot;:

Square root of 56045.8276

There must be an even number of digits
both to the right of the decimal and
also to the left of the decimal. since there
are 5 digits to the left of the decimal, and
5 is odd, so we annex a 0 in front.
                ___________
               &lt;font face = &quot;symbol&quot;&gt;Ö&lt;/font&gt;056045.8276

Pair the digits by skipping spaces like this

              __ __ __.__ __
             &lt;font face = &quot;symbol&quot;&gt;Ö&lt;/font&gt;05 60 45.82 76

Over the 05 put the largest digit whose
square does not exceed 5. This is 2.

 
              &lt;u&gt; 2&lt;/u&gt; &lt;u&gt;  &lt;/u&gt; &lt;u&gt;  &lt;/u&gt;.&lt;u&gt;  &lt;/u&gt; &lt;u&gt;  &lt;/u&gt;
             &lt;font face = &quot;symbol&quot;&gt;Ö&lt;/font&gt;05 60 45.82 76 

Square 2, getting 4. Put 4 under the 5

 
              &lt;u&gt; 2&lt;/u&gt; &lt;u&gt;  &lt;/u&gt; &lt;u&gt;  &lt;/u&gt;.&lt;u&gt;  &lt;/u&gt; &lt;u&gt;  &lt;/u&gt;
             &lt;font face = &quot;symbol&quot;&gt;Ö&lt;/font&gt;05 60 45.82 76 
               4

Subtract, get 1.

              &lt;u&gt; 2&lt;/u&gt; &lt;u&gt;  &lt;/u&gt; &lt;u&gt;  &lt;/u&gt;.&lt;u&gt;  &lt;/u&gt; &lt;u&gt;  &lt;/u&gt;
             &lt;font face = &quot;symbol&quot;&gt;Ö&lt;/font&gt;05 60 45.82 76 
               &lt;u&gt;4&lt;/u&gt;
               1

Bring down the next two digits, 60

              &lt;u&gt; 2&lt;/u&gt; &lt;u&gt;  &lt;/u&gt; &lt;u&gt;  &lt;/u&gt;.&lt;u&gt;  &lt;/u&gt; &lt;u&gt;  &lt;/u&gt;
             &lt;font face = &quot;symbol&quot;&gt;Ö&lt;/font&gt;05 60 45.82 76 
               &lt;u&gt;4&lt;/u&gt;
               1 60

Multiply the 2 on top by 20 to get the
trial divisor 40. Draw a vertical line and
write 40 to the left of 1 60

              &lt;u&gt; 2&lt;/u&gt; &lt;u&gt;  &lt;/u&gt; &lt;u&gt;  &lt;/u&gt;.&lt;u&gt;  &lt;/u&gt; &lt;u&gt;  &lt;/u&gt;
             &lt;font face = &quot;symbol&quot;&gt;Ö&lt;/font&gt;05 60 45.82 76 
               &lt;u&gt;4&lt;/u&gt;
            40|1 60

40 goes into 160 4 times. So try 4 for the
next digit, but add 4 to the trial divisor
40 before multiplying by 4, making it 44


              &lt;u&gt; 2&lt;/u&gt; &lt;u&gt; 4&lt;/u&gt; &lt;u&gt;  &lt;/u&gt;.&lt;u&gt;  &lt;/u&gt; &lt;u&gt;  &lt;/u&gt;
             &lt;font face = &quot;symbol&quot;&gt;Ö&lt;/font&gt;05 60 45.82 76 
               &lt;u&gt;4&lt;/u&gt;
            40|1 60
            44| 

Multiply the 4 by the 44, getting 176
Put that at the bottom
              &lt;u&gt; 2&lt;/u&gt; &lt;u&gt; 4&lt;/u&gt; &lt;u&gt;  &lt;/u&gt;.&lt;u&gt;  &lt;/u&gt; &lt;u&gt;  &lt;/u&gt;
             &lt;font face = &quot;symbol&quot;&gt;Ö&lt;/font&gt;05 60 45.82 76 
               &lt;u&gt;4&lt;/u&gt;
            40|1 60
            44|1 76 

Oh oh! 176 is larger than 160, So the 4 was
too big, so erase the 4, 44, and the 176, and
try 1 less than 4, which is 3. 

              &lt;u&gt; 2&lt;/u&gt; &lt;u&gt; 3&lt;/u&gt; &lt;u&gt;  &lt;/u&gt;.&lt;u&gt;  &lt;/u&gt; &lt;u&gt;  &lt;/u&gt;
             &lt;font face = &quot;symbol&quot;&gt;Ö&lt;/font&gt;05 60 45.82 76 
               &lt;u&gt;4&lt;/u&gt;
            40|1 60
            43|

Multiply the 3 by the 43, getting 129
and write it at the bottom

              &lt;u&gt; 2&lt;/u&gt; &lt;u&gt; 3&lt;/u&gt; &lt;u&gt;  &lt;/u&gt;.&lt;u&gt;  &lt;/u&gt; &lt;u&gt;  &lt;/u&gt;
             &lt;font face = &quot;symbol&quot;&gt;Ö&lt;/font&gt;05 60 45.82 76 
               &lt;u&gt;4&lt;/u&gt;
            40|1 60
            43|1 29


So we subtract and bring down the next two digits

              &lt;u&gt; 2&lt;/u&gt; &lt;u&gt; 3&lt;/u&gt; &lt;u&gt;  &lt;/u&gt;.&lt;u&gt;  &lt;/u&gt; &lt;u&gt;  &lt;/u&gt;
             &lt;font face = &quot;symbol&quot;&gt;Ö&lt;/font&gt;05 60 45.82 76 
               &lt;u&gt;4&lt;/u&gt;
            40|1 60
            43|&lt;u&gt;1 29&lt;/u&gt;
                 31 45             

Multiply the 23 on top by 20 to get the
trial divisor 460. Draw a vertical line and
write 460 to the left of 31 45

              &lt;u&gt; 2&lt;/u&gt; &lt;u&gt; 3&lt;/u&gt; &lt;u&gt;  &lt;/u&gt;.&lt;u&gt;  &lt;/u&gt; &lt;u&gt;  &lt;/u&gt;
             &lt;font face = &quot;symbol&quot;&gt;Ö&lt;/font&gt;05 60 45.82 76 
               &lt;u&gt;4&lt;/u&gt;
            40|1 60
            43|&lt;u&gt;1 29&lt;/u&gt;
             460|31 45

460 goes into 3145 6 times. So try 6 for the
next digit, but add it to the trial divisor
460 before multiplying by 6, making it 466

              &lt;u&gt; 2&lt;/u&gt; &lt;u&gt; 3&lt;/u&gt; &lt;u&gt; 6&lt;/u&gt;.&lt;u&gt;  &lt;/u&gt; &lt;u&gt;  &lt;/u&gt;
             &lt;font face = &quot;symbol&quot;&gt;Ö&lt;/font&gt;05 60 45.82 76 
               &lt;u&gt;4&lt;/u&gt;
            40|1 60
            43|&lt;u&gt;1 29&lt;/u&gt;
             460|31 45
             466|

Multiply the 6 by the 466, getting 2796
and write it at the bottom


              &lt;u&gt; 2&lt;/u&gt; &lt;u&gt; 3&lt;/u&gt; &lt;u&gt; 6&lt;/u&gt;.&lt;u&gt;  &lt;/u&gt; &lt;u&gt;  &lt;/u&gt;
             &lt;font face = &quot;symbol&quot;&gt;Ö&lt;/font&gt;05 60 45.82 76 
               &lt;u&gt;4&lt;/u&gt;
            40|1 60
            43|&lt;u&gt;1 29&lt;/u&gt;
             460|31 45
             466|27 96

So we subtract and bring down the next two digits

              &lt;u&gt; 2&lt;/u&gt; &lt;u&gt; 3&lt;/u&gt; &lt;u&gt; 6&lt;/u&gt;.&lt;u&gt;  &lt;/u&gt; &lt;u&gt;  &lt;/u&gt;
             &lt;font face = &quot;symbol&quot;&gt;Ö&lt;/font&gt;05 60 45.82 76 
               &lt;u&gt;4&lt;/u&gt;
            40|1 60
            43|&lt;u&gt;1 29&lt;/u&gt;
             460|31 45
             466|&lt;u&gt;27 96&lt;/u&gt;
                  3 49 82


Multiply the 236 on top by 20 to get the
trial divisor 4720. Draw a vertical line and
write 4720 to the left of 3 49 82

              &lt;u&gt; 2&lt;/u&gt; &lt;u&gt; 3&lt;/u&gt; &lt;u&gt; 6&lt;/u&gt;.&lt;u&gt;  &lt;/u&gt; &lt;u&gt;  &lt;/u&gt;
             &lt;font face = &quot;symbol&quot;&gt;Ö&lt;/font&gt;05 60 45.82 76 
               &lt;u&gt;4&lt;/u&gt;
            40|1 60
            43|&lt;u&gt;1 29&lt;/u&gt;
             460|31 45
             466|&lt;u&gt;27 96&lt;/u&gt;
             4720|3 49 82

4720 goes into 34982 7 times. So try 7 for the
next digit, but add 7 to the trial divisor
4720 before multiplying by 7, making it 4727

              &lt;u&gt; 2&lt;/u&gt; &lt;u&gt; 3&lt;/u&gt; &lt;u&gt; 6&lt;/u&gt;.&lt;u&gt; 7&lt;/u&gt; &lt;u&gt;  &lt;/u&gt;
             &lt;font face = &quot;symbol&quot;&gt;Ö&lt;/font&gt;05 60 45.82 76 
               &lt;u&gt;4&lt;/u&gt;
            40|1 60
            43|&lt;u&gt;1 29&lt;/u&gt;
             460|31 45
             466|&lt;u&gt;27 96&lt;/u&gt;
             4720|3 49 82
             4727|


Multiply the 7 by the 4727, getting 33089
and write it at the bottom


              &lt;u&gt; 2&lt;/u&gt; &lt;u&gt; 3&lt;/u&gt; &lt;u&gt; 6&lt;/u&gt;.&lt;u&gt; 7&lt;/u&gt; &lt;u&gt;  &lt;/u&gt;
             &lt;font face = &quot;symbol&quot;&gt;Ö&lt;/font&gt;05 60 45.82 76 
               &lt;u&gt;4&lt;/u&gt;
            40|1 60
            43|&lt;u&gt;1 29&lt;/u&gt;
             460|31 45
             466|&lt;u&gt;27 96&lt;/u&gt;
             4720|3 49 82
             4727|3 30 89


Subtract and bring down the next
two digits:

              &lt;u&gt; 2&lt;/u&gt; &lt;u&gt; 3&lt;/u&gt; &lt;u&gt; 6&lt;/u&gt;.&lt;u&gt; 7&lt;/u&gt; &lt;u&gt;  &lt;/u&gt;
             &lt;font face = &quot;symbol&quot;&gt;Ö&lt;/font&gt;05 60 45.82 76 
               &lt;u&gt;4&lt;/u&gt;
            40|1 60
            43|&lt;u&gt;1 29&lt;/u&gt;
             460|31 45
             466|&lt;u&gt;27 96&lt;/u&gt;
             4720|3 49 82
             4727|&lt;u&gt;3 30 89&lt;/u&gt;
                    18 93 76                      

Multiply the 2357 (ignoring the decimal) on
top by 20 to get the trial divisor 47140. 
Draw a vertical line and
write 47140 to the left of 18 95 76


              &lt;u&gt; 2&lt;/u&gt; &lt;u&gt; 3&lt;/u&gt; &lt;u&gt; 6&lt;/u&gt;.&lt;u&gt; 7&lt;/u&gt; &lt;u&gt;  &lt;/u&gt;
             &lt;font face = &quot;symbol&quot;&gt;Ö&lt;/font&gt;05 60 45.82 76 
               &lt;u&gt;4&lt;/u&gt;
            40|1 60
            43|&lt;u&gt;1 29&lt;/u&gt;
             460|31 45
             466|&lt;u&gt;27 96&lt;/u&gt;
             4720|3 49 82
             4727|&lt;u&gt;3 30 89&lt;/u&gt;
              47140|18 93 76 

47140 goes into 189376 4 times. So try 4 for the
next digit, but add 4 to the trial divisor
47140 before multiplying by 4, making it 47144

              &lt;u&gt; 2&lt;/u&gt; &lt;u&gt; 3&lt;/u&gt; &lt;u&gt; 6&lt;/u&gt;.&lt;u&gt; 7&lt;/u&gt; &lt;u&gt; 4&lt;/u&gt;
             &lt;font face = &quot;symbol&quot;&gt;Ö&lt;/font&gt;05 60 45.82 76 
               &lt;u&gt;4&lt;/u&gt;
            40|1 60
            43|&lt;u&gt;1 29&lt;/u&gt;
             460|31 45
             466|&lt;u&gt;27 96&lt;/u&gt;
             4720|3 49 82
             4727|&lt;u&gt;3 30 89&lt;/u&gt;
              47340|18 93 76 
              47344|

Multiply the 4 by the 47344, getting 189376
Put that at the bottom.

              &lt;u&gt; 2&lt;/u&gt; &lt;u&gt; 3&lt;/u&gt; &lt;u&gt; 6&lt;/u&gt;.&lt;u&gt; 7&lt;/u&gt; &lt;u&gt; 4&lt;/u&gt;
             &lt;font face = &quot;symbol&quot;&gt;Ö&lt;/font&gt;05 60 45.82 76 
               &lt;u&gt;4&lt;/u&gt;
            40|1 60
            43|&lt;u&gt;1 29&lt;/u&gt;
             460|31 45
             466|&lt;u&gt;27 96&lt;/u&gt;
             4720|3 49 82
             4727|&lt;u&gt;3 30 89&lt;/u&gt;
              47340|18 93 76 
              47344|&lt;u&gt;18 93 76&lt;/u&gt;

Subtract and get 0 remainder.

              &lt;u&gt; 2&lt;/u&gt; &lt;u&gt; 3&lt;/u&gt; &lt;u&gt; 6&lt;/u&gt;.&lt;u&gt; 7&lt;/u&gt; &lt;u&gt; 4&lt;/u&gt; 
             &lt;font face = &quot;symbol&quot;&gt;Ö&lt;/font&gt;05 60 45.82 76 
               &lt;u&gt;4&lt;/u&gt;
            40|1 60
            43|&lt;u&gt;1 29&lt;/u&gt;
             460|31 45
             466|&lt;u&gt;27 96&lt;/u&gt;
             4720|3 49 82
             4727|&lt;u&gt;3 30 89&lt;/u&gt;
              47340|18 93 76 
              47344|&lt;u&gt;18 93 76&lt;/u&gt;
                           0

If you hadn&#39;t gotten 0 for remainder then
you would annex 2 more 0&#39;s and continue
on with the same pattern.

Edwin&lt;/pre&gt;       </PRE>