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# Discipline Roots

## Extracting roots quickly

Try to solve the following task by mental calculation:

Extracting a root is a complex arithmetic problem. Only a very small number of persons are able to solve this without use of a calculator. The mental math masters are able to solve even these problems mentally.

If someone is talking about extracting roots, they usually mean the square root. The root of two is called the square root. A root is the inverse operation of a square. If we take the 2nd power of 9, it means nine squared or 9*9 = 81. If we want to extract the root of 81, it is again 9. This is the second power, or the square root. Herewith you can find an overview of the powers from 1-32.

If you look at the results, you can see that the powers of 1 and 9 always end with 1. The powers of 2 and 8 always end with 4, the powers of 3 and 7 end with 9 and the powers of 4 & 6 always end with 6. The powers of 5 end with 5, the power of x0 ends with 0. This is important for the further procedure.

### In school, we have learned to extract roots in the following way:

• 1. Divide the number to the left in duads
• 2. Deduct unequal numbers from the left group. Start with 1 and continue until there is one positive rest left. This means: 7-1=6, 6-3=3, 3-5= - 2 is not possible.
• 3. Count the number of unequal numbers, This brings us to the first figure of the solution(2).
• 4. Add the next duad (50) to the (positive) rest (3). This brings us to the number 350.
• 5. Multiply the current result by 2 (2x2=4). This is the new basis to which we add the unequal numbers (4X) and deduct them from the value (350).
• 6. Follow the same procedure as indicated under item 2 : 350-41=309, 309-43=266, 266-45…..​‍‌‍‍‌‍‌‍​
• 7. Follow the same procedures as indicated under items 3. -5. 3 : Number of unequal numbers (7), 2. number of the solution, 4 : Add the next duad (2176), 5. Multiply the result by 2(27x2 = 54) ​‍‌‍‍‌‍‍‌​
• 8. Continue as indicated under item 4. Rest (21) and next duad (76), this makes (2176). 2176-541=1635, 1635-543=1092,… ​‍‌‍‍‌‍‍‍​
The steps from item 5 on can be repeated until the result is sufficiently accurate or the remainder is 0.

### Another way to extract square roots:

To do so, you need the powers mentioned at the beginning of the article.
• We take the sample example : extract the square root of 75076
• We divide the number 75076 in two blocs : 750 & 76
• The power therefore ends with 76. This brings us to the powers 4 or 6 because they always end with 6.
• Now we are looking for the biggest possible power not exceeding 750. This brings us to 27 because 27x27=729.
• Therefore, the solution can only be 27 4 or 27 6.
• We use a trick and take the power in between ending with 5, namely the power of 275.
• It is relatively easy to calculate powers ending with 5. To do so, we split 275 in 27 & 5. Then we take 5x5 = 25 and 27 x 27+1 = 27x28. This brings us to the power of 27 = 729+27= 756. Now you have to combine both results : 75625.
• Now you have to check which option applies: If the root we are looking for (75076) is lower than the power of xx5 (75625), the lower power applies, if the root is higher, the higher power applies.
• The power we are looking for is lower than the power ending with 5, 274 is therefore the right solution.

• Let me illustrate the whole procedure by taking another example : We are looking for the square root of 12769. We can split the number in two blocs : 127 & 69. The smaller bloc ends with 9. The powers we have to take into consideration are the powers from 3 and 9. The highest possible power not exceeding 127 is 11. This brings us to the following options : 11 3 & 11 9. Now we have to take the power of 115. 11x11 = 121.+ 11= 132. 132 & 25= 13225. The number lies above 12769. The root we are looking for is 12769=113.

### This is the way to do it be mental calculation:

Until now, I could not find reasonable instructions illustrating how to extract square roots. Therefore, I am still waiting for the instructions of a mental math master.

## Instructions: Extractings roots: How to calculate a square root

The task is to extract the square root of a five-digit number mentally. With little training you will be able to achieve this goal. How to extract roots quickly is described under Root extraction method.

The more you improve, the higher your level.

### These are the different levels:

110101010
220202020
330303030
440604040
550905050
6601206060
7611236161
8711537171
9811838182
109121391102
11101243101122
12111273111142
13121303121162
14122306122164
15132336132184
16142366142204
17152396152224
18162426162244
19172456172264
20200540200320