Square 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=