Algebra 1
Simplifying Radicals
Prentice Hall Algebra 1 section 10-2
Name______________________________
1. For the following expressions, find the missing values:
12 =____
22 = ____
32 = ____
42 = ____
52 = ____
62 = ____
72 =____
82 = ____
92 = ____
102 = ____
112 = ____
122 = ____
132 =____
142 = ____
152 = ____
162 = ____
202 = ____
252 = ____
2. One definition of a perfect square is “a number that when the square root is taken, the
result is a whole number (no decimals)”. Examples of perfect squares are: 1, 4, 9, 16, etc.
a. List below five more perfect squares.
b. When you take the square root of a perfect square you get a whole number. Simplify the
following:
1=
€
4=
81 =
− 121 =
€ any number that when
€ multiplied together
€ form a product. For example, 3 is
3. A factor is
a factor of 12, because 3 can be multiplied by 4 to give 12. Similarly, 4 is a factor of 40,
because 4 times 10 is 40.
a. List all 6 factors of 18.
b. Cross out the 1, we will not be including 1 in our lists of factors in this investigation because
every number has a factor of 1, and 1 = 1 .
c. From the remaining factors, circle the one that is a perfect square in red pencil.
€
4. Below each number, list all the factors of that number (excluding 1) and circle the
perfect squares in red pencil.
75
45
200
For Algebra 1, a simplified radical is a square root that contains no perfect squares factors
under the radical sign (besides 1). In mathematics the square root symbol is called a radical
sign. The number under the radical sign is called the radicand.
For instance, 3 18 isn’t in simplified radical form because you know that 18 has a perfect
square factor, 9. (Look back at #3) So if there is a perfect square factor (other than 1)
under the radical sign it is NOT simplified.
€ is a list of radical expressions. The number in front of the radical is being
5. Here
multiplied by the radical, just like a coefficient. Below each expression list the factors of
the radicand (the number under the square root symbol).
3 25
€
€
19
2 32
4 35
€
7 2
€
8
3 36
€
5 18
a Circle in red all factors that are perfect squares.
€ perfect square factors
€ that did not have any
€ are already simplified. Underline
b. All expressions
these in green.
6. In mathematics we like to have simplified forms of numbers so that we can compare
them. It can be frustrating if people write the same answer in many different forms.
Just like writing fractions in their simplest form makes them easier to compare, writing
radical expressions in their simplest form makes comparisons easier.
a. For example, look at 3 25 . We know the square root of 25 is 5. So this expression means
3• 5 . It is a lot simpler to write 15 instead of 3 25 . In the same way simplify 2 36 .
€
b. Some of the€others can be simplified as well. For instance look at 7 16 • 3 . This is the
same thing as 7 16 • 3 . Circle the€part of this expression that can be€simplified. What do
you think the simplified form will be?
€
7. One€thing you just learned is that when you have a product under a radical sign we can
break it apart into two radicals and it is still true.
a. For example
4 •9= 4 • 9
36 = ____ • _____ Simplify the square roots of 4 and 9.
€
€
_____=_________
€
€
b. So in general
€
€
a • b = ___ • ___
€
Simplify both sides.
8. Listed below are the factors of 20 (excluding 1).
2
4
5
10
20
a. Circle with red pencil the one that is a perfect square.
b. If I was trying to simplify
perfect square.
20 =
€
20 , I could rewrite it as a product, using the factor that is a
4 •5
The reason I would do this is because it can be simplified.
€
€
We know
20 =
4 •5 =
4• 5
4 is 2. So if we replace
4 with 2, the simplified form of
20 = 2 5
€
€
€
9. Listed below are the factors of 90.
€ 2 45
•
3 • 30€
€
5 • 18
9 •€10
a. Circle the factor pair you would choose if we were going to put
€
b. Rewrite
90 in simplified form.
€
€
€
90 into the factors you chose. Then break it apart into two square roots.
90 =
____• ____ =
€
___ • ___
€
€
€ you
€ will be able to simplify one of the square roots. Do that now. If
c. If€you’ve chosen well,
you can’t, then go back to #10a and choose the factor pair that has a perfect square.
10. Simplify the following radical expressions. You will need to:
First – find a perfect square factor of the radicand (the number under the radical).
Second – rewrite the radicand as a product of two factors (one is a perfect square).
Third – break the radical into two square roots.
Fourth – simplify the radical that has a perfect square.
28
€
50
€
63
€
24
€
242
€
11. Analyze the following equivalent radicals. Underline the two that are ready to be
simplified, cross out the others.
48
€
16 • 3
€
2 • 24
€
4 •12
€
6•8
€
a. Why did you choose the two you underlined?
b. Break those you underlined into two square roots each and simplify the part you can.
c. Of the two you simplified circle the one is in simplified radical form. How do you know?
12. Write below the one you didn’t circle from #11c.
a. What is the number under the radical (the radicand)?______
b. List all the factors of that number.
c. Is there a perfect square among those factors?
d. See if you can simplify it further.
13. The following radical expressions have been simplified a bit. But they haven’t been
simplified all the way. Simplify them until they have no perfect square factors in the
radicand.
2 80
€
7 72
€
11 300
€
3 242
€
14.
Open to a blank page of your toolkit, label the page “Simplifying Radicals”. Add all
underlined vocabulary as well as a few examples.