Instant Mental Calculation of Square Roots





Square Root Calculation

Square Root Calculation

Learn how to mentally calculate square roots and amaze your friends!


Calculating square roots is rather more difficult than either cube roots or fifth roots - you might want to learn these two methods first.


Following a few simple steps you will be able to instantly calculate the square root of the spectator's number.

To perform this, you should ask the spectator to choose any whole number less than 100 and, using a calculator, to square it by multiplying the number by itself.

The spectator then calls out the answer, and you instantly reveal the original number (i.e., the square root).


Note that this method will NOT work with non-whole numbers, or with numbers greater than 99.


Step 1: Learn the Squares of 0 to 9


To master the system you must learn by heart the squares of numbers 0 to 9, which are shown in the table below. You also need to consider the last digit of each square.


NumberSquareLast Digit
000
111
244
399
4166
5255
6366
7499
8644
9811


Note how the last digits for the squares of 1 and 9 are both 1.

Note how the last digits for the squares of 2 and 8 are both 4.

Note how the last digits for the squares of 3 and 7 are both 9.

Note how the last digits for the squares of 4 and 6 are both 6.


Step 2: Determine the First Digit

Ignore the last TWO digits of the number called out by the spectator and choose the memorized comparison square which is just lower (or equal) to the remaining number. The corresponding square root is the first digit. of your answer


Example 1

Spectator calls 676

  • Ignore 76, leaving 6
  • The comparison square which is just lower than 6 is 4
  • Therefore the first digit of the answer is the square root of 4 = 2

Example 2

Spectator calls 5184

  • Ignore 84, leaving 51
  • The comparison square which is just lower than 51 is 49
  • Therefore the first digit of the answer is the square root of 49 = 7

Step 3: Determine the Second Digit


Now consider the last digit of the number called out by the spectator and compare this to the last digit of the memorized squares.

  • If the last digit of the spectator's call is 0, then the last digit of your answer is also 0.
  • If the last digit of the spectator's call is 5, then the last digit of your answer is 5.
  • In all other cases, the last digit of the spectator's call will indicate two possible values for the last digit of the square root. For example, if the last digit of the spectator's call is 9, then the square root may end in either 3 or 7.

To determine whether the lower or higher of two possible values should be taken, multiply the first digit of your answer (as found in Step 2) by one greater than itself.

  • If this is greater than (i.e., ignoring the last two digits), then the last digit of your answer is the lower of the two possible values.
  • If this is less than (or equal to) the first part of the number called by the spectator, then the last digit of your answer is the higher value.

Example 1

Spectator calls 676

  • Last digit of spectator's call = 6
  • Therefore the two possible values of the second digit are 4 or 6
  • First digit of answer (from Step 2) is 2
  • Multiply 2 by (2+1) = 6
  • Because 6 is equal to the first part of the spectator's call (6), then the second digit will be the higher of the two possible values
  • Therefore the second digit is 6 (not 4)
  • Therefore the square root of 676 is 26

Example 2

Spectator calls 5184

  • Last digit of spectator's call = 4
  • Therefore the two possible values of the second digit are 2 or 8
  • First digit of answer (from Step 2) is 7
  • Multiply 7 by (7+1) = 56
  • Because 56 is greater than the first part of the spectator's call (51), then the second digit will be the lower of the two possible values
  • Therefore the second digit is 2 (not 8)
  • Therefore the square root of 5184 is 72

Further Examples


Call
SQ
1st
Part
(P)
Lower
(or =)
SQ
1st
Digit
(F)
2nd
Digit?
F x
(F+1)
Compare
with P
2nd
Digit
(S)
SQRT
= FS
1441112 or 822 > 1212
7847422 or 866 < 7828
136913933 or 71212 < 13737
2025201645  545
2401241641 or 92020 < 24949
2809282553 or 73030 > 28353
3600363660 060
5041504971 or 95656 > 50171
7396736484 or 67272 < 73686
8836888194 or 69090 > 88494

Practice, Practice, Practice!


Before you try this out on your friends, you should practice until you can calculate the square root instantly and without error.

If you can't do this, then practice some more!

To practice and assess your ability, you can use the Square Root Tester shown below.


Square Root Tester



 

 
 

 




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