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Algebra II – Chapter 6 Day #1
Topic: Roots and Radical Expressions
Standards/Goals:
 A.SSE.2: I can take the structure of an expression and identify different ways to rewrite it
 G.1.b.: I can simplify radicals that have various indices.
 G.1.c.: I can use properties of roots and rational exponents to simplify expressions.
Today, we will focus on introducing ourselves to some new aspects to square roots and radicals that
we have not encountered before.
Here are some quiz things to consider when considering a radical:
𝑛
√𝑎
 √ is known as the ‘radical sign’
 ‘a’ is known as the ‘radicand’
 ‘n’ is known as the ‘index’ OR ‘root’
o Some things to know about: 𝒂𝒏 = 𝒃
 If ‘n’ is odd, then there is ONE real root
 If ‘n’ is even and positive, then there are TWO real roots. The positive root
is what we call the ‘principal’ root.
 If ‘n’ is even and negative, then there are NO real roots.
 The only ‘nth’ root of 0 is 0.
We first want to be able find all the ‘real’ roots of a number.
EXAMPLES:
What are the real CUBE roots of the following numbers?
27
#1. -64
#2. 0.001
#3. 125
What are all the real square roots of the following numbers?
25
#4. -0.81
#5.
#6. 121
What are the real fourth roots of the following numbers?
#7. 1
#8. -0.00001
#9.
What are the real fifth roots of the following numbers?
#10. 0
#11. -1
#12. 32
49
16
81
2
In this chapter, we are only focusing on ‘real’ roots. In the past, we focused on imaginary and complex
roots. According to the Fundamental Theorem of Algebra, 𝑥 4 − 1 = 0, has only four roots, only two of
which are real. In a problem like this, let’s only focus on the real roots only.
We now want to find the real roots of numbers.
What is each real number root of the following numbers?
3
#3. √(−5)2
3
#6. √(−17)2
4
#9. √0.04
#1. √0.16
#2. √−343
#4. √0.36
#5. √−1
3
#8. √−1
#7. √−8
We also want to be able to simplify radical expressions when they have variables present.
Before, we learn how to simplify radical expressions, let’s go over an important property that is
important to consider.
PROPERTY: for ‘nth’ Roots of ‘nth’ Powers
𝑎, 𝑖𝑓 𝑛 𝑖𝑠 𝑜𝑑𝑑
𝑛
For any real numbers a, √𝑎𝑛 = {
|𝑎| 𝑖𝑓 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛
What is the simplified form of each radical expression?
4
#3. √16𝑥 8
3
#6. √𝑥 8 𝑦12
3
#9. √𝑥12 𝑦16
#1. √25𝑎4
#2. √𝑝12 𝑞 4
#4. √9𝑥 2
#5. √𝑥 6 𝑦15
3
#8. √𝑎12 𝑏15
#7. √𝑎6 𝑏 9
4
4
3
HOMEWORK – Chapter 6 Day #1
Name ________________________________ Date _____
Find all the real square roots of each number.
1. 400
2. 196
3. 10,000
4. 0.0625
Find all the real cube roots of each number.
6. 343
5. 216
7. 0.064
8.
1000
27
Find all the real fourth roots of each number.
9. 81
10. 256
11. 0.0001
12. 625
Find each real root.
13. 144
14.  25
16. 3 0.001
17. 4 0.0081
18.
3
15. 0.01
19.
27
3
27
20. 0.09
Simplify each radical expression. Use absolute value symbols when needed.
81x 4
21.
25.
5
22.
243x5 y15
121y10
26.
3
23.
0.008
3
8g 6
24.
4
30.
x4
81
3
125x 9
31.
36x 2 y 6