HANDOUT
LAWS OF EXPONENTS
Name
: ________________________________
Grade/Number
: ____________________
A. Index Notation / Exponents
Index notation is a short way of writing repeated multiplication
𝑎𝑛 = 𝑎 × 𝑎 × 𝑎 × … × 𝑎
𝑛
where,
𝑎𝑛 is called an index notation with a positive integer power. It is read as 𝑎 to the power of
𝑛
𝑎 is called a base, 𝑎 ∈ 𝑅
𝑛 is called the power of the index notation, 𝑛 ∈ Ζ+
B. Laws of Exponents
Let 𝑎, 𝑏 ∈ 𝑅, and 𝑚, 𝑛 are integers, with 𝑚 ≥ 𝑛, then :
Law
𝑎1 = 𝑎
𝑎0 = 1
1
𝑎−1 =
𝑎
𝑎𝑚 . 𝑎𝑛 = 𝑎𝑚+𝑛
Example
61 = 6
70 = 1
1
4−1 =
4
𝑎2 . 𝑎3 = 𝑎2+3 = 𝑎5
𝑎𝑚
= 𝑎𝑚−𝑛
𝑎𝑛
𝑎6
= 𝑎6−2 = 𝑎4
𝑎2
(𝑎𝑚 )𝒏 = 𝑎𝑚𝑛
(𝑎2 )3 = 𝑎2.3 = 𝑎6
(𝑎𝑏)𝑚 = 𝑎𝑚 𝑏𝑚
(𝑎𝑏)2 = 𝑎2 𝑏 2
𝑎 𝑚 𝑎𝑚
( ) = 𝑚
𝑏
𝑏
𝑎 3 𝑎3
( ) = 3
𝑏
𝑏
𝑎−𝑛 =
𝑚
𝒏
1
𝑎𝑛
𝑎 𝑛 = √𝑎𝑚
𝑎−3 =
2
1
𝑎3
3
𝑎 3 = √𝑎 2
C. Examples
1. Express the following numbers in positive powers and simplify :
a. (𝑝3 𝑞 −2 ) × (𝑝−5 𝑞−1 )
−2
b.
(3𝑝−2 𝑞3 )
(32 𝑝−1 𝑞2 )−3
2. Simplify the following expressions and write the results in positive integers.
a.
5𝑥 2 𝑦 4 ×4𝑥 5 𝑦
22 𝑥 2 𝑦 2
b.
𝑝−7 𝑞−1 𝑟 10
(−𝑝𝑞)5
∶
𝑝2 (−𝑞)2
𝑞𝑟 −3
3. It is known that the population in an area is 50.000 peoples in 2020. Experts have
observed that the population growth rate in the area is about 2.5% per year.
a. Write an equation for the total population in the area
b. What will be the population in 2050?
D. EXERCISES
1. Simplify the following expressions and write the results in positive power forms.
a.
b.
3𝑥 3 𝑦 5 ×10𝑥𝑦 4
5𝑥 2 𝑦 6
−1
3𝑎 6 𝑏−5
(81𝑎9 𝑏−1 )
24𝑎3 𝑏7
6𝑏3 𝑎
2
c. ( 8𝑎5 𝑏2 ) × ( 3𝑎3 )
2. Gavriel saves in a bank. Gavriel's initial savings was IDR 50,000,000.00. Every month
Gavriel gets additional money with a percentage of 2% per month.
a. Write the equation of Gavriel's savings in the bank
b. How much does Gavriel save after 5 years?
3. An entomologist monitors the presence of swarms of insects that will invade an area.
The formula for the monitored area is expressed by 𝐴(𝑛) = 1000 × 50,5𝑛 , where 𝑛 is the
number of weeks since the monitoring was carried out. If in the past few weeks the
area affected by insects is 5,000 hectares, determine the fastest time for the insects to
attack.