9
Learning Activity Sheet 2:
Zero, Negative and Rational
Exponents
Mary Joy E. Aban, Irene D. Iglesias
Writers
Engr. Aurora A. Quiambao
Evaluator
Lesson 1: ZERO AND NEGATIVE EXPONENTS
Learning Competency: Applies the laws involving positive integral exponents to zero and negative integral
exponents
Objectives: Apply the laws involving positive integral exponents to zero and negative integral exponents
Reference: Learner’s Materials for Grade 9
Grade 9 Mathematics Patterns and Practicalities
E-Math Worktext in Mathematics 9
LAWS OF EXPONENTS
If a and b are real numbers and m and n are positive integers, then
The Product Rule: 𝒂𝒎 • 𝒂𝒏 = 𝒂𝒎+𝒏
Example: 52 • 23 = 52+3 = 55 = 3 125
(𝑥 + 𝑦)3 (𝑥 + 𝑦)4 = (𝑥 + 𝑦)3+4 = (𝑥 + 𝑦)7
𝒂𝒎
The Quotient Rule: = 𝒂𝒎−𝒏 Any number raised to the power of
𝒂𝒏
53 one equals the number itself. 𝑎1 = 𝑎
Example: = 53−2 = 51 = 5
52
Raising a Power to a Power Rule: (𝒂𝒎 )𝒏 = 𝒂𝒎𝒏
Example: (53 )2 = 5(3)(2) = 56 = 15 625
Raising a Product to a Power Rule: (𝒂𝒃)𝒎 = 𝒂𝒎 𝒃𝒎
Example: (𝑎𝑏)𝑚 = [(5)(3)]3 = (53 ) (33 ) = (125)(27) = 3 375
𝒂 𝒎 𝒂𝒎
Raising a Quotient to a Power Rule: ( ) =
𝒃 𝒃𝒎
3 32 9
Example: ( 4)2 = 4 2 =16
Zero Exponent:
If a is a nonzero number, then 𝑎 0 = 1
Example:
1. 52 •50 = 25 • 1 = 25
2. (30 )2 = (1)2 = 1
24 24
3. = = 24 = 16
20 1
4. [(2)(3)]0 = (20 )(30 ) = (1)(1)=1
2 0 20 1
5. ( ) = =1=1
4 40
6. (5𝑥 3 𝑦 2 𝑧)0 = 50 𝑥 3(0)𝑦 2(0) 𝑧 0 = 1(𝑥 0 )(𝑦 0 )(1) = 1(1)(1)(1) = 1
7. (𝑥 2 − 𝑦 2 )0 = 1
45
8. = 45−5 = 40 = 1
45
Negative Exponents
1
If n is a positive integer and if a is a nonzero real number, 𝑎 −𝑛 =
𝑎𝑛
−𝑛 𝑛
The number 𝑎 and 𝑎 are reciprocals. Thus, whenever you come across a negative
exponent, think “reciprocal”.
1 1
1. 5−2 = 52 = 25
1 1 1 1 1
2. 2−4 • 5−2 = 24 • 52 = 16 • 25 = 400
32 1 1
3. = 32−5 = 3−3 = 33 = 27
35
1 2 1 2 1 2 12 1
4. (6−2 )2 = (62) = (62) = (36) = 362
= 1296
3 −2
5. ( )
2
=
1 1
3 −2 3−2 32 9 1 4 4
Method 1: ( ) = −2 = 1 = 1 = • =9
2 2 9 1
22 4
3 −2 3−2 22 4
Method 2: ( 2) = = 32 = 9
2−2
Drills:
Simplify each expression.
(𝑥+2)−9
1. 𝑥 0 • 𝑥 −5 • 𝑥 4 = 6. =
(𝑥+2)−5
10−5
2. (3−2 )2 = 7. =
106
𝑥 −9𝑥 4
3. (𝑎−2 𝑏)5 = 8. =
𝑥5
4. (𝑎 + 𝑏 + 𝑐 )0 = 9. (−2𝑥 −5 𝑦 3 )−4 =
(𝑥+𝑦)−2 (𝑥+𝑦)4
5. 43 •40 • 45 = 10. =
(𝑥+𝑦)3
Lesson 2: RATIONAL EXPONENTS
Learning Competency: Simplify expressions with rational exponents; and
Write expressions with rational exponents as radicals and vice versa.
Reference: Learner’s Materials for Grade 9
Objectives: Illustrate expressions with rational exponents,
Simplify expressions with rational exponents; and
Write expressions with rational exponents as radicals and vice versa.
1
𝑛 𝑛 𝑛
For any integer n > 1 if any real number 𝑎 for which √𝑎 is defined, 𝑎 𝑛 = √𝑎 . In √𝑎 , 𝑎 is the
radicand and 𝑛 is the index of the radical.
𝑚
𝑛
Examples: 𝑥 𝑛 = √𝑥 𝑚
1
22 = √2
𝐍 𝐭𝐡 𝐑𝐨𝐨𝐭 𝐨𝐟 𝒂
𝑛 𝑛
The 𝑛𝑡ℎ root of a is denoted by √𝑎 , and √𝑎 = 𝑏 if 𝑏 𝑛 = 𝑎.
If n is an even natural number, a must be positive or zero and b must be positive.
Example 1:
1
a. 92 = √9 = 3
1
3
b. (8)3 = √64 = 4
1
c. 362 = √25 = 5
1
5
d. (−1024)5 = √−1024 = -4
3 4 3 4 9+8 15 5
e. (𝑦 2 ) (𝑦 3 ) = 𝑦 2+3 = 𝑦 6 = 𝑦 6 = 𝑦2
1
3
f. 273 = √27 = 3 Since 3 (3)(3) = 33 = 27
4
g. √−81 is not a real number since (−3)4 ≠ −81 and (3)4 ≠ −81.
Remember:
When the index n is an even
𝑛
number, √−𝑎 is not a real number, but
𝑛
when n is odd, √−𝑎 is real number.
Example 2: Product Rule 𝑎 𝑚 • 𝑎 𝑛 = 𝑎 𝑚+𝑛
1 1 1 1 2 1
+
a. 164 • 164 = 164 4 = 164 or 162 = √16 =4
1 1 2+1 3 1
b. 9 •9 = 9
3 6 6 = 9 = 9 = √9 = 3
6 2
1 1 1 1
c. 52 • 5−2 = 52+(−2) = 50 = 1
𝑎𝑚
Example 3: Quotient Rule = 𝑎 𝑚−𝑛
𝑎𝑛
Note: When dividing exponential expressions with the same bases, find the difference of the exponents.
𝑎 𝑚 𝑎𝑚
Example 4: Quotient to a Power Rule ( ) = 𝑏𝑚
𝑏
3 3−2 1
−
625 4 4
a. 1 = 625 4 = 6254 = √625 = 5
6252
1
1 1
195 − 0
b. 1 = 195 5 =19 = 1
195
2
2 1 1
273 − 3
1 = 27
c. 3 3 = 273 = √27 = 3
273
Example 5: Raising a Power to a Power Rule : (𝑎 𝑚 )𝑛 = 𝑎 𝑚𝑛
1 3 3
a. (73 ) = 73 = 71 = 7
1 2 2 2
3 3 3
b. (8 ) = 83 = √82 = √64 = 4 or ( √8) = 22
3
1 3 3
c. (5 ) = 52 = √53 = √125 = 5√5
2
Example 6: Product to a power Rule : (𝑎𝑏 )𝑚 = 𝑎 𝑚 𝑏 𝑚
1 1 1
•
a. (9 • 16)2 = 92 162 = √9 •√16 = 3•4 =12
1
1442 = √144 = 12
1 1 1
3 3
b. (27 • 8)3 = 273 •83 = √27• √8 = 3 • 2 = 6
𝒂 𝒎 𝒂𝒎
Example 7: Quotient to a Power Rule: ( ) = 𝒎
𝒃 𝒃
1 1
3
27 3 273 √27 3
a. ( ) = 1 =3 =
64 √64 4
643
1 1
9 2 92 √9 3
b. ( ) = 1 = =
16 √16 4
162
Negative Rational Exponent
If 𝑚 𝑎𝑛𝑑 𝑛 are positive integers that are relatively prime, where 𝑎 is a real number and 𝑎 ≠ 0 , then
−𝑚 2
1 𝑛
𝑎 𝑛 =( 𝑚 ) = provided √𝑎 is a real number.
𝑎𝑛
2 2
1 1 12 1 1 1 1
a. (8−3 ) = ( 1 ) = 1 = 2 =3 2
=3 =4
83
( )(2)
8 3 83 √8 √64
1 2 1 2 12 1
= ( 3 ) = ( 2) = 22 = 4
√8
1 1 1 1 −2−1 −3 1
1 1 1
b. 9 −
3 • 9−6 = 9−3+(−6) = 9 6 = 9 6 = 9−2 = 1 = =
9 3
92 √
Writing expression with Rational exponents to Radical and vice versa
1
3
(𝑦 2 +2)3 √𝑦 2 +2
1. 1 = 1 = 3√𝑦 2 + 2 • 3√𝑦 2 − 2 = 3√𝑦 4 − 4
(𝑦 2 −2)−3 3 2
√𝑦 −2
3
2. 4𝑏 = √43 𝑏 3 = √64𝑏 3 = 8√𝑏 3
2
3
4 4 4 4 4 4
3. √8𝑝 9 = √23 (𝑝3 )3 = √(2𝑝 3 ) = (2𝑝 3 )4 = √23 𝑝3(3) = √23 𝑝 9 = √8𝑝9
1
1
4. 5 = 2−5
√2
1 1
3
3 9 9 3 93 √9 3 9
5. √ = (10) = 1 = 3 = √10
10 √10
103
Drills:
I. Simplify the following rational exponents:
1 1
1. 22 • 2−2 =
5
256
2. 1 =
253
1 2
42
3. ( ) =3
44
1
32 5
4. ( )=
243
1
5. (25 • 4)2 =
II. Write expressions with rational exponents as radicals and vice versa:
2
1. 3𝑛5 =
1
2 3
2. ( 2) =
3𝑥
1
3. 5 2
2
4. 5 3
6
5. √64𝑦 5
Assessment:
Choose the letter of the best answer and write it on a separate sheet of paper.
1
1. What is the simplified form of 90 3−2 5−1 750 5−1 ?
1 1 1
a. b. c. d. 1
75 81 9
2. Which of the following is true?
1 1 1 1
−3
a. 22 + 23 = 25 c. 2 = 1
23
1
1 2 32 2
b. (4 ) = 16 2 d. 1 = 39
33
−1 3
3. Simplify the expression 4𝑥 𝑦 .
𝑦3 4𝑦 3 1
a. b. c. 4𝑦 3 d.
4𝑥 𝑥 4𝑥𝑦 3
4. Evaluate −43 • 32 • 70
a. −576 b. 4 032 c. 84 d. 504
𝑝𝑚
5. What value of 𝑚 will make the statement true in = 1?
𝑝6
a. 12 b. 0 c. 1 d. 6
𝑛 −𝑛
6. Simplify 225 • 225
1
a. 15 b. 0 c. d. 1
15
7. Which of the following statements is always true? Assume that 𝑐, 𝑑 𝑎𝑛𝑑 𝑚 are positive integers.
1
a. 𝑐𝑑 = 𝑑𝑐 b. 𝑐 −𝑑 = −𝑑 𝑐 c. 𝑐 −𝑑 = d. 𝑐 𝑚 = 𝑑 𝑚
𝑐𝑑
8. What value of 𝑘 will make 7𝑘−4 −23 = 41?
a. 6 b. 4 c. 1 d. 5
1 1
9. Multiply 144 • 144 4 4
1
a. 120 b. 12 c. 10 d.
4
3 −1
2435
10. Divide ( 4 )
2435
a. 2 b. −1 c. 0 d. 3
5 3
11. What is equivalent of √3 + √9 using exponential notation?
1 1 1
a. 35 + 9 3 b . 3 5 + 93 c. 128 d. 128
1 1
12. What is the radical form of 5𝑎 2 + 2𝑏 2 ?
a. 5√2𝑎 + √2𝑏 b. √5𝑎 + √2𝑏 c. 5√𝑎 + 2√𝑏 d. √5𝑎 + 2𝑏
3 2
364
13. Simplify ( 1 )
362
1 1
a. b. 6 c. d. -6
36 2
1
14. Simplify (1024) 5
a. 24 b. 5 c. 4 d. 10
1
15. What value of 𝑥 will make (81) = 3 𝑥
a. 2 b. 3 c. 4 d. 5
