RADICAL 9

1. RADICALS

2. 25
52
25=
= 25−52
= 5

3. RADICAL
• It is an expression consisting of
a radical sign and radicand.
• Its type is 𝒏
𝒂, where n is the
index and a is the radicand.
• When no index is indicated, the
radical sign indicates a square root.

4. 23
2 • 2 • 2
𝟑
𝟖
8= Radical
sign
Radicand
Index
Radical
The number 2
is the cube
root of 8.

5. EXAMPLES:
1. 49 = 7
Indicates the principal
square root of 49.
2. 36 = 6
Indicates the principal
square root of 36.
3. 2 2 is not a perfect square
but the value of its square
root can be approximated.

6. EXAMPLES:
1. 9 =
Indicates the principal
square root of 9.
2. 100 =
Indicates the principal
square root of 100.
3
10

7. EXAMPLES:
1. 25 = 5
3. 64 =
2. 16 = 4
8

8. EXAMPLES:
1.
3
64 = 4
4 is used as a factor three times. 4 is
also considered the principal cube root.
2.
4
81 = 3
3 is used as a factor four times. 3 is
also considered as the principal 4th
root of 81.
3.
5
−32 = −2 If the index is odd, we can get the
principal root of any real number.
4.
3
5
5 is not a perfect cube so we cannot get
the exact cube root. We can only get its
approximate value.

9. EXAMPLES:
1.
3
8 =
2. 100 =
2
10

10. RATIONAL
NUMBER
Examples:
6 =
6
1
, 0 =
0
1
, 4 =
4
1
It is a number
which can be
expressed as a
ratio of two
integers.

11. IRRATIONAL
NUMBER
It is a number
which cannot
be expressed
as a ratio of
two integers.
Examples:
2,
3
5,
π

12. REMEMBER THAT:
• The cube root of positive number is positive.
• The cube root of negative number is negative.
• There is only one real number cube root for each
real number.
• When the index is an even number ( like square
root, fourth root, and so on) and the radicand is
positive, there exists a principal root.

13. Tell which number is the index and
which is the radicand.
• 4
•
3
8
•
5
12
•
10
7
•
48
12
•
5
4
•
9
3
•
5
29
• 27
• 49
• 216
•
3
−125
•
3 144
96
•
3
1000

14. QUIZ
Evaluate the following radicals.

15. ZERO
and
NEGATIVE
EXPONENTS

16. Rule:
x0
= 1
Where x ≠ 0.
ZERO
EXPONENTS
= 𝑎5−5
E 𝑥𝑎𝑚𝑝𝑙𝑒:
𝑎5
𝑎5
= 𝑎0
= 1
𝑎 ≠ 0

17. EXAMPLES:
1. 80=
2.(
32
22)0
=
1
1
3. 80
+ 80
= 2
4. 82
• 80
= 64

18. Rule:
𝒙−𝒏
=
𝟏
𝒙 𝒏 and 𝒙 𝒏
=
𝟏
𝒙−𝒏
Where 𝒙 ≠ 0 and n is a
counting number.
NEGATIVE
EXPONENTS
=
𝟏
𝟓 𝟑
E 𝑥𝑎𝑚𝑝𝑙𝑒:
𝟓−𝟑
=
𝟏
𝟏𝟐𝟓

19. EXAMPLES:
1. 8−2
=
2.( 2−2
)−3
=
𝟏
𝟔𝟒
𝟔𝟒
3.(
1
2
)−2 =
𝟏
𝟑
4. 3 • 3−3
=
4
1
82
=
𝟏
(
𝟏
𝟐
) 𝟐
=
𝟏
(
𝟏
𝟐
) 𝟐
=
3 •
1
33
=

20. ASSIGNMENT
To be
submitted on
SEPTEMBER
20, 2020
(SUNDAY)
until 4pm.

21. ASSIGNMENT
1.
3
64 =
2. 1 =
3. 4 =
4. 49=
5.
3
216 =
4
6
2
7
1
6. 27=
7.
3
−8 =
8. 81 =
9. 144 =
10. 64 =
32 • 4 = 3 4 𝒐𝒓 5.20
-2
9
12
8

22. ASSIGNMENT
To be submitted on
SEPTEMBER 20,
2020 (SUNDAY)
until 4pm.
To be submitted on
SEPTEMBER 20,
2020 (SUNDAY)
until 4pm.

23. ASSIGNMENT
1. 1
2. 1
3.
1
𝑥 𝑛
4. 𝑥−𝑛

24. ASSIGNMENT
1.
1
7
or
2.
1
53
3.
1
81
4.
1
121
5. 1
6.5,764,801
7.
1
𝑎4
8. 3
9. 1
10. 7
11. 1
12. 4,096
13. 9
14.
5
3
15.1

25. RATIONAL
EXPONENTS

26. Which one
is a
rational
exponents?
• 16
1
2
•
3
27
• 8
1
2
• 16
1
4
• 27
1
3
• 2
1
2
• 1243
1
5
•
3
56
•
3
256
• 27
•
3
81

27. Rational
Exponents
For any integer
n > 1 any real
number b for which
𝒏
𝒃 is defined, 𝒃
𝟏
𝒏 =
𝒏
𝒃.

28. EXAMPLES:
1. 𝟗
𝟏
𝟐 =
2. (𝟔𝟒)
𝟏
𝟑
3
4
3. 𝟐𝟓
𝟏
𝟐
𝟗 =
𝟑
𝟔𝟒 =
𝟐𝟓 = 5
4. −(𝟐𝟒𝟑)
𝟏
𝟓
−(
𝟓
𝟐𝟒𝟑) = -3

29. QUIZ In your notebook answer the following.

30. Writing the
Equivalent
Exponential
Expression
of a Radical
Expression
• Write the radicand as the
base of the exponential
expression.
• The reciprocal of the
index becomes the
exponent.

31. POWER LAW
OF
EXPONENTS
If m and n are positive integers
with no common factor except
1, then for all real numbers b for
which 𝒃
𝟏
𝒏 is defined,
𝒃
𝒎
𝒏 = (𝒃
𝟏
𝒏) 𝒎
= (
𝒏
𝒃 )
𝒎
or (
𝒏
𝒃 𝒎 )

32. EXAMPLES:
1. 𝟐𝟕
𝟐
𝟑 =
2. 𝟏𝟔
𝟑
𝟒 =
8
3. −𝟐𝟓
𝟑
𝟐 =
(
𝟒
𝟏𝟔) 𝟑
=
-125
(𝟑) 𝟐
=(
𝟑
𝟐𝟕) 𝟐
=
(𝟐) 𝟑=
9
−( 𝟐𝟓) 𝟑
=
−(𝟓) 𝟑
=

33. ASSIGNMENT
In your notebook answer the following.

34. ASSIGNMENT In your notebook answer
the following.

35. ASSIGNMENT
In your notebook answer the following.