How to estimate the square root of a number
that is not a perfect square
• The square root of a number
is a value that, when
multiplied by itself, gives the
number.
• When finding the square root
of a number the square root
symbol is used to denote the
principal (positive) square
root. The square root symbol
is called a radical and the
number under the radical is
called the radicand.
• Essentially taking the square
root of a number is the
inverse operation or opposite
of squaring a number.
square
3
radical
square
root
9
radicand
Principal
Square
root
9= 3
Any positive number has two square
roots – a positive and a negative.
2
(+6)
(+6) · (+6) or
= +36
2
(−6) · (−6) or (−6) = +36
So 36 = +6 𝑎𝑛𝑑 − 6
This is the same as ±6
Now you try … What are the two
square roots of 64?
We know that
2
(+8)
(+8) · (+8) or
= +64
2
(−8) · (−8) or (−8) = +64
So 64 = +8 𝑎𝑛𝑑 − 8
This is the same as ±8
Important Note:
When finding the square root
of a number you are usually
asked to find the principal
square root or non-negative
root of that number unless
otherwise denoted.
You know to find the negative square root of a number
when the negative sign is on the outside of the radical.
− 49 = −7
Now you try ….
− 81 = −9
Be mindful though that the −25 is not a real
number. The reason is because there are not two
identical real numbers that when multiplied
together results in -25.
5 x 5 is positive 25 and -5 x -5 is positive 25.
−25 is considered “not a real number” and
taking the square root of -25 results in an
“imaginary” number.
The explanation of “imaginary” numbers is
beyond the scope of this podcast.
So it is easy to find the square
root of a perfect square that
is an integer but what about
square roots such as 30, in
which 30 (the radicand) is not
a perfect square?
First, in order to understand how to estimate square roots of
non-perfect squares, or a square root of a number that does not
result in an integer, let’s first list the perfect squares and their
roots.
Number
Squared
Perfect
Square
After
taking the
square
root the
result is
12 22 32 42 52 62 72 82 92 102 112 122 132 142 152
1
4
9 16 25 36 49 64 81 100 121 144 169 196 225
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15
Next, figure out which two integers that
30 is between.
25 = 5
We see that
30 = ? 36 = 6
30 is between the integers 5 and 6.
This means that
30 is 5.?
• But what does the 30 exactly equal?
• 30 is not a perfect square, so there are no two identical
factors that when multiplied together will result in 30.
• So when taking the 30 it will not result in a an integer
(whole number or the negative of a whole number).
• However, we can drill down to find the two identical
factors that when multiplied together will give you a
number close to 30.
• Usually when estimating the square root of a number
that is not a perfect square such as 30 without a
calculator, estimating to the tenths or hundreths
decimal place is sufficient.
Let’s try it.
We know that the square root of 30 is “5 point something”.
Next, square 5.1, 5.2 and so forth and place it in a table.
Numbered
squared
5.12
5.22
5.32
5.42
5.52
5.62
5.72
5.82
5.92
Result
26.01
27.04
28.09
29.16
30.25
31.36
32.49
33.64
34.81
From the table we see
that 5.42 results in 29.16
and 5.52 results in 30.25.
So the square root of 30
falls between 5.4 and 5.5.
We can drill down even more to the hundreths place by
creating another table.
Number
Squared
5.412
5.422
5.432
5.442
5.452
5.462
5.472
5.482
5.492
Result
29.2681
29.3764
29.4849
29.5936
29.7025
29.8116
29.9209
30.0304
30.1401
From here we can see
that 5.472 and 5.482
results in a number
closest to 30.
But which is the closest?
We can figure this out by subtracting.
5.472
29.9209
30 – 29.9209
.0791
5.482
30.0304
30.0304 - 30
.0304
So for the square root of 30, estimating to the hundredths place
we see that 5.48 is the best estimate because the difference is the
smallest.
We can repeat this process and drill down further to the
thousandths, ten thousandths place and so on. We’d be here
forever because there is no terminating or repeating decimal that
when multiplied by itself results in 30.
We can check our result of 5.48 on a calculator, and we get
5.477225575051661…
5.4772… rounds up to 5.48