This isn't necessarily a question to be answered. But more like "lost knowledge" that I can now do. Of all of my grades K-12 and college courses up to Calculus, not a single teacher or professor taught anyone how to do square roots and cube roots or even nth roots by hand. Although nth roots are tricky starting at the 4th root, you can cheat it by doing a double square root. Even my textbooks where showing how to do it in a calculator. And I feel that a calculator shouldn't be dependent on forever. Because technology will ultimately dumb us down if taken too far. For now, I will show you how to do square roots and cube roots by hand.

Lets start with the square root of 144 for example. Something simple.

Code:
  _____
\|144
Step 1) Underline the numbers in double digits beginning from the right to the left. If there is one digit left, then underline that one digit.

Code:
_____
|144
 - --
Step 2) Next, ask yourself this. What two numbers of the same number multiplied together will be or get as close to 1 as possible without overshooting it?

1² = 1
So the answer is 1.


Put a 1 over the square root. And subtract the 1 digit from the 144 by 1. Then drop the next 2 digits. If no digits are available and its not zero after the subtraction, drop 2 0's and add a decimal.

Code:
_1___
|144
-1 vv
-----
 044
Step 3) Since you got 1², instead of multiplying by 1*1, add 1+1 to get 2, but leave a blank space after 2 to get 2_.

Code:
    _1___
2_|144
    -1 vv
    ----
     044
Step 4) Now ask yourself. What 2_ times its own missing digit will be or get as close to 44 as possible without overshooting it?

20 * 0 = 0
21 * 1 = 21
[22 * 2 = 44]

Step 5) Since you got 2, put it on top, and subtract it by 44.

Code:
    _12__
22|144
    -1 vv
    -----
     044
     - 44
    -----
        0
And that is it! Now if it were a bigger number or a decimal, you would keep adding blank digits. Like if we had to keep going, we would drop 2 0's, add 22 + 2 to make 24, and put a blank next to 24_. Now ask yourself. What 24_ times its own missing digit will be or get as close to 0 as possible without overshooting it? 0! We can't go no further. If you really wanna test this technique, try any number. Hell try even the square root of 2. It'll never end, but rest assure the number will match in your calculator

Now on to cube roots. This is a little tricky, as there are a few more steps involved. But nothing we can't handle. Lets try the cube root of 91125

Code:
3 _____
\|91125
I used this number so you can see the technique first hand.

Step 1) Underline the numbers in triple digits beginning from the right to the left. If there is one digit left, then underline that one digit. If there are two, underline the two digits.

Code:
______
|91125
  -- ---
Step 2) Next, ask yourself this. What three numbers of the same number multiplied together will be or get as close to 91 as possible without overshooting it?

1³ = 1
2³ = 8
3³ = 27
[4³ = 64]
5³ = 125

So the answer is 4.

Put a 4 over the cube root. And subtract the 91 digits from 91125 by 64. Then drop the next 3 digits. If no digits are available and its not zero after the subtraction, drop 3 0's and add a decimal.

Code:
_4___
|91125
-64 vvv
-----
 27125
Step 3) Since you got 4³, instead of multiplying by 4*4*4, add 4+4+4 to get 12, but leave a blank space after 12 to get 12_.

Code:
      _4___
12_|91125
     -64 vvv
      -----
      27125
Step 4) Now multiplying the 12_ by its own digit 3 times will not work necessarily. There is another trick involved here. Were gonna involve the first 3 digits of our final number, which is 271, and divide it by the 12 we obtained times the 4 we obtained to get an approximation rounded down. We only need the whole number from this:

271/(12*4) ≈ 5


Step 5) With this number you have obtained, you will multiply it by the 4 you obtained earlier squared, multiplied by 300.

5 * 4² * 300 = 24000

Step 6) Plug in the missing value for 12_ with 5 and multiply by 5 twice.

125 * 5 * 5 = 3125

Step 7) Add em up

24000 + 3125 = 27125

Step 8) Subtract it from the total, and put the 5 above the cube root. It became 0. Thats it!

Code:
      _45__
125|91125
     -64 vvv
      -----
      27125
     -27125
     --------
             0
The final answer is 45. I honestly don't understand why they don't teach this in elementary or middle school cause it was so damn easy to figure out. And the technique is dying due to a high abundance of calculators. Even the textbooks are telling you to use a calculator! Let me know what you guys thing.