Radicals chapter notes

1. 1
CLO 2.2: Rational Exponents
LSM 0103 – APPLIED MATH FUNDAMENTALS
Aims and Objectives
1. Simplifying nth roots
2. Simplifying 𝑎
1
𝑛 and 𝑎
𝑚
𝑛
3. Simplifying expressions with rational exponents
Reference:
Connect with eText: College Algebra, Miller/Gerken,
Section R.3 Pages 32-35
Pages 37-38: #53-94
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2. 2
2.2.1. Simplifying nth roots
For example:
36 = 36
1
2 = ? 𝑂𝑅 ? ? = 36
3
125 = 125
1
3 = ? 𝑂𝑅 ? ? ? = 125
4
81 = 81
1
4 = ? 𝑂𝑅 ? ? ? ? = 81
Example 1: Simplify:
a.
5
32 b.
49
64
c.
3
−8 d.
4
−1 e. −
4
1

3. 3
2.2.2. Simplifying 𝑎
1
𝑛 and 𝑎
𝑚
𝑛
Example 2: Write the expressions using radical notation and simplify if possible.
a. 25
1
2 b.
64
27
1
3
c. −81
1
4 d. 323 5 e. −27 2 3

4. 4
2.2.2. Simplifying 𝑎
1
𝑛 and 𝑎
𝑚
𝑛
Exercise 1: Write the expressions using radical notation. Assume that all variables represent
positive real numbers.
a. 𝑦
4
11 b. 6𝑦
4
11 c. 6𝑦
4
11
Exercise 2: Write the expressions using rational exponents. Assume that all variables represent
positive real numbers.
a.
5
𝑎3 b.
7
𝑧4 c. 6𝑥 c. 6 𝑥

5. 5
2.2.3. Simplifying expressions with rational exponents
Example 3: Simplify. Assume that all variables represent
positive real numbers.
a.
𝑥
4
7 𝑥
2
7
𝑥
1
7
b. 81−
3
4

6. 6
2.2.3. Simplifying expressions with rational exponents
Exercise 4: Simplify each expression. Assume that all variables represent positive real numbers.
a.
𝑎
2
3 𝑎
5
3
𝑎
1
3
b.
𝑦
7
5 𝑦
4
5
𝑦
1
5
c.
3𝑤
−2
3
𝑦
−1
3

7. 7
7
CLO 2.2: Simplifying Radical
LSM 0103 – APPLIED MATH FUNDAMENTALS
Aims and Objectives
2.2.4. Review Basic Properties of Radicals
2.2.5. Simplify Radicals Using the Product Property
2.2.6. Add and Subtract Radicals
Reference:
Connect with eText: College Algebra, Miller/Gerken,
Section R.3 Pages 32-35
Examples 4, 5, 6, 7
Practice: Pages 37-38: #53-94
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8. 8
2.2.4. Review Basic Properties of Radicals
Example 1: Simplify:
a)
3
53 b) −4 2 c)
3
32 ∙
3
2 d)
5
−7 5

9. 9
2.2.4. Review Basic Properties of Radicals
Example 2: Write as a product of prime numbers:
a. 50 b. 24 c. 54
Prime factorization to Simplify Radicals
Prime Numbers are the numbers that
have only two factors, 1 and the number
itself. For example, 2, 3, 5, 7, 11, 13, 17,
19, and so on are prime numbers.
Prime Factorization of any number
means to represent that number as a
product of prime numbers.
Tree Method:
Step 1: Place the number on top of the
factor tree.
Step 2: Then, write down the
corresponding pair of factors as the
branches of the tree.
Step 3: Factorize the composite factors
that are found in step 2, and write down
the pair of factors as the next branches of
the tree.
Step 4: Repeat step 3, until we get the
prime factors of all the composite factors.
Write 40 as a product of prime numbers.
40 = 2 × 2 × 2 × 5

10. 10
2.2.4. Review Basic Properties of Radicals
Example 3: Simplify:
3
24𝑥6𝑦4
Step 1: 3
2 ∙ 2 ∙ 2 ∙ 3 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑦 ∙ 𝑦 ∙ 𝑦 ∙ 𝑦
Step 2: Notice the index of the radical is 3. Create groups of 3.
3
2 ∙ 2 ∙ 2 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑦 ∙ 𝑦 ∙ 𝑦 ∙ 3 ∙ 𝑦
Step 3:
3
23 ∙
3
𝑥3 ∙
3
𝑥3 ∙
3
𝑦3 ∙ 3
3 ∙ 𝑦
Step 4: 2 ∙ 𝑥 ∙ 𝑥 ∙ 𝑦 ∙ 3
3 ∙ 𝑦
= 2𝑥2
𝑦 ∙ 3
3𝑦
Steps to Simplifying Higher order
Radicals
Step 1: Find the prime factorization of
the number inside the radical and factor
each variable inside the radical.
Step 2: Determine the index of the
radical and group the numbers and
variables according to that index.
Step 3: For each group, rewrite them as
𝑛
𝑥𝑛, leaving the remaining factors under
the radical at the end.
Step 4: Simplify by removing radicals
and multiplying as needed.

11. 11
2.2.4. Review Basic Properties of Radicals
Example 4: Simplify: 20𝑥3𝑦7𝑧2
Step 1:
Step 2:
Step 3:
Step 4:
Steps to Simplifying Higher order Radicals
Step 1: Find the prime factorization of the
number inside the radical and factor each
variable inside the radical.
Step 2: Determine the index of the radical and
group the numbers and variables according to
that index.
Step 3: For each group, rewrite them as
𝑛
𝑥𝑛,
leaving the remaining factors under the radical
at the end.
Step 4: Simplify by removing radicals and
multiplying as needed.

12. 12
2.2.5. Simplify Radicals Using the Product Property
Example 5: Simplify each expression. Assume all variables represent positive real numbers.

13. 13
2.2.5. Simplify Radicals Using the Product Property
Practice: Simplify each expression. Assume all variables represent positive real numbers.

14. 14
2.2.5. Simplify Radicals Using the Product Property
Exercises: Simplify each expression. Assume all variables represent positive real numbers.
1.
3
40𝑎𝑏3𝑐7 2.
4
96𝑝6𝑞8

15. 15
2.2.6. Add and Subtract Radicals
Example 6: Add or subtract as indicated. Assume that all
variables represent positive real numbers.
a) 5
3
7𝑡2 − 2
3
7𝑡2 +
3
7𝑡2 b) 3 5𝑥 + 2 5𝑥
Remember
You add or subtract
radicals like variables
𝒙 + 𝟐𝒙 = 𝟑𝒙
𝟓 + 𝟐 𝟓 = 𝟑 𝟓

16. 16
2.2.6. Add and Subtract Radicals
Practice: Add or subtract as indicated. Assume that all variables represent positive real numbers.
a) −4
3
5𝑤 + 9
3
5𝑤 − 11
3
5𝑤 b) 8 7𝑧 + 3 7𝑧

17. 17
Exercises:
For Exercises 1 and 2, simplify each expression. Assume that all variable expressions represent
positive real numbers.
1.
3
54𝑎𝑏6𝑐8 2.
5
32𝑝7𝑞10

18. 18
Exercises:
For Exercises 3 and 4, simplify each expression. Assume that all variable expressions represent
positive real numbers.
1. 2.

19. 19
Exercises:
For Exercises 5 and 6, add or subtract as indicated. Assume that all variable expressions represent
positive real numbers.

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