1. 9.1 Properties of Radicals

2. What We Will Learn Use properties of radicals to simplify
Perform operations with radicals

3. Essential Question
How can you multiply and divide square roots?

4. Needed Vocab Radical expression: expression that contains a radical
Simplest form: cannot be broken down any further
Like radicals: same index and radicand

5. Ex. 1 Using Product Property
108
36 × 3
6 3 24
4 × 6
2 6 75
25 × 3
5 3
128
64 × 2
8 2
Steps
1. Find biggest perfect square that divides into number with no decimal
2. Separate 𝑟𝑎𝑑𝑖𝑐𝑎𝑛𝑑 into perfect square and number left after dividing
𝑝𝑒𝑟𝑓𝑒𝑐𝑡 𝑠𝑞𝑢𝑎𝑟𝑒 ∗ 𝑟𝑒𝑚𝑎𝑖𝑛𝑑𝑒𝑟
3. Square root perfect square
4. leave 𝑟𝑒𝑚𝑎𝑖𝑛𝑑𝑒𝑟 alone

6. Ex. 1 Continue Simplify Letters
𝑎 6
𝑎 3
𝑎 6 𝑏 12
𝑎 3 𝑏 6
𝑐 11
𝑐 10 × 𝑐 1
𝑐 5 𝑐
𝑑 15 𝑒 9
𝑑 14 × 𝑑 1 × 𝑒 8 × 𝑒 1
𝑑 7 𝑒 4 𝑑𝑒
𝑎 10 𝑏 11
Leave evens alone
𝑎 10 × 𝑏 10 × 𝑏 1
𝑎 5 𝑏 5 𝑏
If exponent is even, then just cut in half
If exponent is odd, then find biggest even number
Split into the even number exponent and exponent of one
Split even number in half, and leave exponent of one alone

7. Ex. 1 Continued Combining Numbers and Letters
108 𝑥 4
36 × 3 × 𝑥 4
6 𝑥 2 3
120 𝑎 6 𝑏 9
4 × 30 × 𝑎 6 × 𝑏 9
4 × 30 × 𝑎 6 × 𝑏 8 × 𝑏 1
2 𝑎 3 𝑏 𝑏
Your Practice
125 𝑛 7 𝑡 9
25 × 5 × 𝑛 6 × 𝑛 1 × 𝑡 8 × 𝑡 1
5 𝑛 3 𝑡 4 5𝑛𝑡
Steps
1. Split numbers first
Use first notes for numbers
2. Split letters second
Use notes for letters
3. Do perfect square and simplify letters

8. Ex. 2 Using Quotient Property
15 64
15 8
25 ×
Steps
1. Square root top and bottom
2. Simplify using example 1 notes
108 𝑥 𝑦 8
108𝑥 𝑦 8
36 × 3 × 𝑥 4 × 𝑥 × 𝑦 8
6𝑥 2 3𝑥 6𝑦 4
𝑥 2 3𝑥 𝑦 4

9. Your Practice 128𝑥 7 64𝑦 6 128𝑥 7 64𝑦 6 64 × 2 × 𝑥 6 × 𝑥 1 64 × 𝑦 6
128𝑥 𝑦 6
128𝑥 𝑦 6
64 × 2 × 𝑥 6 × 𝑥 × 𝑦 6
8𝑥 3 2𝑥 8𝑦 3
𝑥 3 2𝑥 𝑦 3

10. Ex. 8 Adding/Subtracting Radicals
Must be like radicals
Same index and radicand
Index – root of radical
Radicand – number inside radical
Like Unlike
3 , , 2 5
4 7𝑥 , 4 7𝑥 , 4 𝑥 or 4 7𝑥 , 5 7𝑥

11. Ex. 8 cont.
−8 7
6 3 𝑥 −2 3 𝑥
4 3 𝑥
10 5 − 20
10 5 − 4 × 5
10 5 −2 5
8 5
4 ×
Steps
1. Make into like radicals if can
Use example 1 notes
2. Combine like radicals
Only add or subtract number in front of
LEAVE RADICAL ALONE!!

12. Ex. 9 Multiplying Radicals
− 75
15 − 375
15 − 25 × 15
15 −5 15
-4 15
× 6
When there is a number in front multiply by the square root of the perfect square
36 6
Steps
1. Make sure same index
2. Multiply numbers in front of radical and numbers inside radical
Only front with front, inside with inside
3. Make like radicals
4. Combine like radicals

13. Your Practice 2 5 2 6𝑥 − 96𝑥 4 30𝑥 −2 480𝑥 4 30𝑥 −2 16 × 30𝑥
𝑥 − 96𝑥
4 30𝑥 −2 480𝑥
4 30𝑥 −2 16 × 30𝑥
4 30𝑥 −8 30𝑥
−4 30𝑥