Extracting Square Roots Mentally

Extracting Square Roots Mentally
Presented by Grey Matters Mental Gym
http://members.cox.net/beagenius/


A square is a number multiplied by itself.
For example, 3 squared (32) is 3 * 3, or 9.
A square root is the number that, when
squared, results in a given number. For
example, the square root of 9 (√9) is 3.


Have a spectator choose any number from
1 to 100, square it with their calculator
and give you the answer.
Imagine being able to ﬁgure the square
root of that number in your head!

Requirements

Calculator
Spectator


To extract square roots mentally, you must
know the ﬁrst 10 squares:
12=1 22=4
32=9 42=16
52=25 62=36
72=49 82=64
92=81 102=100


Note that 1, 4, 6 and 9 each appear twice
as the last digit of the squares:
12=1 22=4
32=9 42=16
52=25 62=36
72=49 82=64
92=81 102=100


Also note that 1, 4, 6 and 9 each appear
once below and once above 5:
12=1 22=4
32=9 42=16
52=25 62=36
72=49 82=64
92=81 102=100


Finally, note that the squares for 5 and 0
end in their respective numbers:
12=1 22=4
32=9 42=16
52=25 62=36
72=49 82=64
92=81 102=100


If you’re given one of the squares you’ve
memorized from this chart, simply give the
square root you’ve remembered.


If you’re given a number that ranges from
100 (102) to 10,000 (1002), you’ll need to
go through the following steps.


First, split the number into two parts, with
the rightmost two digits in one part, and
the rest in the other. As an example, let’s
say you’re given the number 1,764. You
would mentally split it into “17” and
“64”.


Next, focusing on the number to the left
set, ask yourself, “What is the largest
square that is equal to or less than that
number?”
In our example, the largest square that is
equal to or less than 17 (the number in the
left set) is 16.


Recall the square root of this number, and
that will be the tens digit of the answer.
In our example, we found that 16 was the
largest square less than or equal to 17.
Since we know that the square root of 16
is 4 (√16=4), we now know our answer is
in the 40s somewhere.


Now, focus on the number in the right set.
In our example of 1,764, this would be the
“64”.


Look at just the rightmost digit, and ask
yourself, “Which digits, from 0-9, when
squared, would result in that digit?”
In our example of 1,764, we see that the
rightmost digit is a 4. However, both 22 (4)
and 82 (64) end in 4, so how do we know
whether 2 or 8 is the right digit?


This is where the trick of squaring
numbers ending in 5 comes in handy.
Using the tens digit discovered earlier, set
5 as the ones digit and square it.
In our “1764” example, we know that the
root is in the 40s, so we square 45, which
is 2,025.


Finally, ask yourself whether the given
number is above or below this square of 5.
This will tell you which digit belongs in
the ones place of the root.
In our example, 1,764 is lower than 2,025
(452), so the lower of the two choices (2,
instead of 8) is correct.


Finally, put the tens digit together with the
ones digit, and you have the answer.
In our example, we determined that 1,764
was in the 40s somewhere, and that 2 was
the correct digit for the ones place, so the
answer is 42!


For squares that end in 5 or 0, you’ll
known that they end in 5 or 0,
respectively.


Other examples:
What is √3,364? Because “33” is greater
than 25 (52), and less than 36 (62), we
know the answer is in the 50s. Because
“64” ends in 4, we know that the answer
must end in 2 or 8. Because 552=3,025,
and 3,364 is greater, √3,364=58.


Other examples:
must end in 5. So, √5,625=75.


Other examples:
must end in 1 or 9. Because 652=4,225,
and 4,761 is greater, √4,761=69.

Extracting Square Roots Mentally

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