Download Given the radical, determine what two perfect squares it is between

GEOMETRY
“An Introduction to Radicals”
Different lengths may be expressed in different ways. We can use whole numbers, decimals, fractions
OR feet, inches, centimeters, or meters to name a few. However certain rules in Geometry use radicals
or square root notation (
). Our objective is for you to learn how to work with radical notation.
As you work through this study guide, there are sections labeled with capital letters. The answers to
these sections are found on a separate sheet that you should occasionally check. Some of the sets we
will go through together.
A square root is a number that can be multiplied by itself to get that number underneath the radical
sign. For example, the square root of 9 is 3 because 3 multiplied by 3 is 9. With radicals, we will only
deal with positive results for now. Know that there IS actually another answer to the above problem:
-3 since –3 multiplied by –3 is also 9.
The first thing that we wish to do is find what are known as the “perfect squares”. We are interested in
the first 20. To find them, take 1  1, 2  2, 3  3, and so on all the way up to 20  20. You can also
think of these as 12, 22, 32 up to 202. We say each of these as “one squared”, two squared”, three
squared”, etc… You will need to know these without a calculator to be successful with radicals!
Write down the first twenty perfect squares in order.
A.
_____, _____, _____, _____, _____, _____, ______, _____, _____, _____
_____, _____, _____, _____, _____, _____, ______, _____, _____, _____
The square root of all whole number perfect squares is a whole number also. Think, what number can
you multiply times itself to get what is underneath the radical sign.
√9 = 3
For example:
√289 = 17
Do the following (your answer will not have the radical sign left):
B.
25
121
100
289
225
36
1
256
144
4
324
81
64
49
9
196
169
400
361
16
A calculator can be used to get a decimal answer for a radical; however, this answer is not exact and is
not always accepted.
Square roots can be estimated by knowing the perfect squares listed above. To determine the square
root of 19, first figure out what perfect squares 19 is between.
19 is between the perfect squares of 16 and 25.
19 is between 4 and 5.
What is your guess? _____
Use a calculator to see how close you are?
Given the radical, determine what two perfect squares it is between. Then without a calculator
estimate the solution to the nearest hundredth. Finally use a calculator to determine the correct
answer to the nearest hundredth.
C.
Radical
Greater Than
This Perfect
Square…
…But Less
than This
Perfect
Square
Estimated
Value
(Hundredths)
Calculated
Value
(Hundredths)
39
11
77
109
62
Decimals are not usually used with perfect squares unless rounding is allowed to take place. You must
learn to simplify just like with other types of numbers. Rules of Geometry are written in radical form.
Let’s simplify square roots where a perfect square can be factored out.
For example:
8
32
15
24
50
1000
Place the following radicals in simplest form:
D.
75
27
72
45
21
300
When taking the square root where a fraction is present and the numerator and the denominator are
perfect squares, one can take the square root of both the top and the bottom separately. Let’s try this
where perfect squares are present:
For example:
4
9
25
36
81
100
64
49
25
169
Place the following radicals in simplest form:
E.
1
4
9
16
9
1
100
121
25
144
You may work with fractions where the denominator is a perfect square and the numerator is not.
Simplify the denominator and leave the numerator as a radical.
For example:
7
4
5
9
10
49
2
25
5
64
Place the following radicals in simplest form:
F.
7
25
3
16
5
81
2
9
11
400
Fractions may have a numerator that is a perfect square but a denominator is not. Often the first thing
to do is to take the square root of the numerator. Then you must rationalize the denominator. This
means that you must eliminate the radical from the bottom, as it is not proper to have a radical in the
denominator. You must multiply the top and the bottom then by the denominator. (You may choose to
simplify the denominator first before rationalizing the denominator…the answer will still be the same.)
For example:
4
7
1
3
9
5
16
5
9
8
25
7
49
12
Place the following radicals in simplest form:
G.
4
3
1
5
1
6
You may have noticed that you can multiply two numbers inside radicals but not a number outside a
radical and a number inside a radical.
Rationalizing must take place even with no perfect squares to be found.
For example:
3
5
5
6
1
7
16
10
2
3
Place the following radicals in simplest form:
H.
3
11
2
7
5
3
3
10
One can also multiply square roots. For example:
14
6
2 3 5 3
8 2 3 6
Find the following products:
I.
5 3 4 5
3 2 5 6
7 3 2 5
Addition and subtraction of square roots can take place only if the numbers underneath the radical are
the same.
For example:
5 3 2 3
6 5 3 5
7 6 7
3 8 2 2
4 5 2 12
Find the following sums and differences:
J. 4 5 2 5
6 12 3 2
5 3 9 3
8 8 3
2 18 5 12