What is the square root of 81

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Rapid root extraction

Try to calculate the following task in your head:
Square root of 75076 =?

Drawing a root is a complex arithmetic task. Very few people can solve this without a calculator. The mental arithmetic masters even do it in their heads.

When someone speaks of taking root, they usually mean the square root. So the second root is called the square root. A root is an inverted power. If we take the 2nd power of 9 that is 9 to the power of 2 or 9 * 9 = 81. If we then want to take the root of 81 that is the 9th again. That is the 2nd power or the square root. As an overview, here are the potencies from 1-32.

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

At school we learned to pull roots like this:

  • 1. Divide the number to the left into groups of two
  • 2. Now subtract odd numbers from the group on the left. Start with 1 until there is a positive remainder! So 7-1 = 6, 6-3 = 3, 3-5 = - 2 no longer works ..
  • 3. Count the number of odd numbers. This is the 1st digit of the solution (2).
  • 4. Add the next group of 2 (50) to the rest (3). That results in the number 350.
  • 5. Multiply the previous result by 2 (2x2 = 4). This is the new base to which we append the odd numbers (4x) and subtract them from the value (350)
  • 6. Proceed as described for 2. 350-41 = 309, 309-43 = 266, 266-45 ... ..
  • 7. Proceed as described under 3. - 5.. 3rd number of odd numbers (7), 2nd digit of the solution. 4. Next group of 2 to this (2176), 5. Result with times 2 (27x2 = 54)
  • 8. Proceed as described from 4. onwards. Remainder (21) and the next block of 2 (76), results in (2176). 2176-541 = 1635, 1635-543 = 1092, ...
You can repeat steps from 5 onwards until the result is sufficiently accurate or the remainder is 0.

Another way to solve a square number:

For this you need the potencies from the beginning of the article.
  • Our example is the root of 75076
  • We divide the 75076 into 2 blocks. 750 & 76
  • So the power ends on 76. This gives the power of 4 & 6, because they always end on 6
  • Now we are looking for the greatest possible potency that does not exceed 750. That is the 27. Because 27x27 = 729
  • So the solution can only be 27 4 or 27 6.
  • We use a trick and take the power that lies in between and ends with 5. So the power of 275.
  • Powers of five are relatively easy to calculate. For this we divide the 275 into 27 & 5. Then we take the 5x5 = 25 and the 27 x 27 + 1 = 27x28. That is then the power of 27 = 729 + 27 = 756. Now connect the two results - 75625
  • Now in the test, the root (75076) you are looking for is below the power of xx5 (75625) then it is the lower of the two possible powers, and if it is above that, it is the other
  • So the one you are looking for is below the power of 5 so the 274 is the right one.

  • Let us take another example for clarification. We are looking for the root of 12769. That results in 2 bucks 127 & 69. So ends on 9. Possible powers with 3 or 9. The greatest power that does not exceed 127 is 11. So possible candidates 11 3 & 11 9. Now that Power of 115. 11x11 = 121. + 11 = 132. 132 & 25 = 13225. That is again above the 12769 we are looking for, ie below the two candidates. The result is root 12769 = 113.

    This is how it goes in the head:

    Since I have not found any reasonable instructions for pulling a square root, I am waiting for instructions from a mental arithmetic athlete.

    Instructions: take the root - calculate the square root

    This is about getting the square root of a five-digit number in your head. With a little practice, you'll be able to do this for sure.

    The better you get, the higher your level gets.

    Here is the level division: