Unit 4 Worksheet 1: Simplifying Radicals
S4.1
I can simplify radical expressions
Terminology

radical is the symbol that represents a square root.

radicand is the number or expression inside a radical symbol


Perfect squares are numbers whose square roots are integers or quotients of integers.
A factor is a number which divides into another number (for example, 5 is a factor of 30).
Properties of Radicals

Product Property
The square root of a product equals the product of the square roots of the factors.
√𝑎𝑏 = √𝑎 ∗ √𝑏 when a and be are positive numbers

Quotient Property
The square root a quotient equals the quotient of the square roots of the numerator and denominator.
𝑎
√𝑏 =
√𝑎
√𝑏
when a and b are positive numbers
Ultimate Goal
 Get radical expressions to their simplest form.
o An expression with radicals is in simplest form if the following are true:
 No perfect square factors other than 1 are in the radicand.
 No fractions are in the radicand.
 No radicals appear in the denominator of the fraction.
Steps to Simplifying a Radical (square root)
1. Check the radicand to see if it is a perfect square.
a. If it is, cross out both the radical and the radicand and replace them with the square root.
2. If the radicand is not a perfect square, look for the largest perfect square that divides into it.
a. 24 is divisible by 4.
72 is divisible by 36.
120 is divisible by 4.
(Notice that while 120 is divisible by many things, like 10 and 12 and 6, the only perfect square that divides into
120 is 4.)
3. Give each number in the product its own radical sign. Always put the perfect square first.
4. Reduce the perfect radical which you have now created.
5. Make sure your radicand does not have any other perfect square factors.
6. You now have your answer!
Perfect Squares
12 = 1
62 = 36
112 = 121
162 = 256
22 = 4
72 = 49
122 = 144
172 = 289
32 = 9
82 = 64
132 = 169
182 = 324
42
92
142
192
= 16
= 81
= 196
= 361
52
102
152
202
= 25
= 100
= 225
= 400
Example
1) √48
√48 = √16 ∗ 3
√48 = √16 ∗ 3 = √16 ∗ √3
√48 = √16 ∗ 3 = √16 ∗ √3 = 4√3
√48 = 4√3
4√3
Step 2: Not a perfect square 
Step 3: 16 ∗ 3 = 48
Step 4
Step 5: √16 = 4
Step 6: √3 has no other perfect squares
Step 7: Finished
Practice
1) √44
2) √27
3) √144
6) √125
7) √200
8)
11) √80
12) √45
13) √72
1
∗
2
√112
4) √75
5) √90
9) √2 ∗ √8
10) 6 ∗ √8
14) √4
15) √24