1. The document provides a review worksheet with problems involving simplifying radical expressions, writing expressions in radical and exponential form, and finding various roots. 2. It includes examples of finding cube roots, square roots, and other roots of numbers, variables, fractions, and decimals. Positive and negative radicands as well as odd and even indexes are addressed. 3. Techniques are presented for rationalizing variables with different exponents and changing decimals to fractions when taking roots.

1. Review: Odd Numbers: Worksheet 1.2 IV. Transform
each into its radical form and simplify as much as
much as possible :
•4.
5
1
2
5
1
3
=
•5. 3
1
2 * 27
1
2=
•6. (210)1/2 =
•7.
29
36
−
1
3
=
•8. 𝑎
1
3𝑎
2
3 =
•9.
𝑎3
𝑏
1
2
−
1
3
=
•1. 8
2
3 =
•2. 16−
3
2
=
•3. 64
3
2 ∗ 64
1
2 =

2. Lesson 3: Square roots
and other roots

3. Square root
• The number “a” is a square root of b if a2=b
• In algebraic form, ± 𝑏 = 𝑎 if and only if a2=b

4. Cube root and other roots
• The number “a” is a nth root of b if an=b
• In algebraic form,
𝑛
𝑏 = 𝑎 if and only if an=b

5. Roots on the calculator:
• https://www.youtube.com/watch?v=ZbocrrjRiR0&t=115s

6. Positive and Negative Radicands
Odd and Even Index
• 1.
3
−8
• 2.
3
8
• 3. 9
• 4. −9
• 5.
4
16
• 6.
4
−16
• 7.
5
32
• 8.
5
−32

7. −1 = 𝑖 𝑖 = 𝐼𝑚𝑎𝑔𝑖𝑛𝑎𝑟𝑦 𝑁𝑢𝑚𝑏𝑒𝑟
Example:
−14 - Think of a number that when you multiply by itself is equal
to -14.
Obviously, there is none.
So we rewrite the given as −1 ∗ 14 where −1 = 𝑖
We now have 𝑖 ∗ 14 𝑜𝑟 14 𝑖

8. 3
−8
Take note that the index is odd.
In the given, think of a number that when you multiply by itself
three times, the answer is equal to -8
−2 −2 −2 = −8
Therefore:
3
−8 =
3
−2 −2 −2 =
3
−2 3 = −2

9. Fractions
To solve the roots of fractions, think of them as two radicals.
Example:
4 81
16
this is the same as
4
81
4
16
4
81 = 3 while
4
16 = 2
Therefore;
4 81
16
=
3
2

10. Fractions
• 1.
4
−
81
16
• 2.
4 81
16
• 3.
3
−
125
27
• 4.
3 125
27

11. Variables
• 1.
3
𝑎3𝑏5𝑐16
• 2.
8
256𝑝24𝑞32𝑧17

12. Variables
To rationalize variables with various exponents, divide the index by the exponent of
each variable, and keep the remainder inside the radical sign.
Example:
3
𝑎3𝑏5𝑐16 - The index is 3
The exponent of a is 3; 3(exponent) divided by 3(index)= no remainder
The exponent of b is 5; 5 divided by 3 = 1 remainder 2
The exponent of c is 16; 16 divided by 3 = 5 remainder 1
Answer:
3
𝑎3𝑏5𝑐16 = 𝑎1
𝑏1
𝑐53
𝑏2𝑐1

13. Decimals
• 1.
𝟑
𝟎. 𝟎𝟎𝟏𝟐𝟓=
• 2.
𝟑
𝟎. 𝟎𝟎𝟎𝟏𝟐𝟓=
• 3.
𝟑
𝟎. 𝟎𝟎𝟎𝟏𝟐𝟓=
For decimals, simply change the decimals to fractions.

14. Let us Practice:
•1. −
3
9
•2. 64
•3.
9
100
•4. 7
•5.
3
−8
• 6. −4
• 7. 0
• 8.
3
0.064
• 9.
3
−1000
• 10.
3 1
64
• 11.
5
−
32
243
• 12.
5
−3125
• 13.
4
−1296
• 14.
3
−512
• 15.
5
−729
• 16.
4 1
81
𝑟16𝑠20

15. Review: Lesson 2-4

16. I. Simplify each. Express your answers using
positive exponents.
• 1. a6b-6b-3
• 2.
𝟑−𝟕
𝟑
−𝟐
∙
−𝟐𝟎
𝟏𝟎𝟎
𝟐

17. II. Write each in radical form or in
exponential form
• 1. 𝐱
𝟑
𝟐 =
• 2.
𝟖
𝟒𝒎 𝟑=

18. III. Find the indicated roots. (Simplify as
much as possible).
• 1.
𝟔
𝒂𝟔𝒃𝟏𝟕𝒄𝟐𝟖=
• 2.
𝟑
𝟎. 𝟏𝟐𝟓=
• 3.
𝟑
𝟎. 𝟎𝟏𝟐𝟓=
• 4.
𝟑
𝟎. 𝟎𝟎𝟏𝟐𝟓=
• 5.
𝟑
𝟎. 𝟎𝟎𝟎𝟏𝟐𝟓=
• 6.
𝟑
𝟎. 𝟎𝟎𝟎𝟏𝟐𝟓=
• 7. 𝟏𝟔 𝟑=
• 8. −𝟒=