1. How to estimate the square root of a number that is not a perfect square
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2. 3 9 9 =3 radical radicand square square root
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 9
square
root
Principal
Square
root
radical
radicand 9 =3

3. Any positive number has two square roots – a positive and a negative
Any positive number has two square roots – a positive and a negative. (+6)·(+6) or (+6) 2 = +36 (−6)·(−6) or (−6) 2 = +36 So 36 = +6 𝑎𝑛𝑑 −6 This is the same as ±6

4. Now you try … What are the two square roots of 64? We know that
(+8)·(+8) or (+8) 2 = +64
(−8)·(−8) or (−8) 2 = +64
So 64 = +8 𝑎𝑛𝑑 −8
This is the same as ±8

5. 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.

6. 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

7. Be mindful though that the −25 is not a real number
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.

8. 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?

9. After taking the square root the result is
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 12 22 32 42 52 62 72 82 92
102
112
122
132
142
152
Perfect Square 1 4 9 16 25 36 49 64 81
100
121
144
169
196
225
After taking the square root the result is 2 3 5 6 7 8 10 11 13 14 15

10. Next, figure out which two integers that 30 is between.
25 =5
30 = ?
36 = 6
We see that is between the integers 5 and 6.
This means that is 5.?

11. 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 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.

12. Let’s try it.

13. So the square root of 30 falls between 5.4 and 5.5.
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
Result
5.12
26.01
5.22
27.04
5.32
28.09
5.42
29.16
5.52
30.25
5.62
31.36
5.72
32.49
5.82
33.64
5.92
34.81
From the table we see that 5.42 results in and 5.52 results in
So the square root of 30 falls between 5.4 and 5.5.

14. We can drill down even more to the hundreths place by creating another table.
Number Squared
Result
5.412
5.422
5.432
5.442
5.452
5.462
5.472
5.482
5.492
From here we can see that and results in a number closest to 30.

15. But which is the closest?
We can figure this out by subtracting.
5.472
30 –
.0791
5.482
.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 … … rounds up to 5.48