Math

1. UNIT I
REAL NUMBER AND REAL LINE

2. 1.1 CONCEPT OF SET
In the different branches of Mathematics and other fields of Science the concept of set
is a basic fundamental notion that we should give importance. The concept of set is so basic and it
is impossible for us to define what is set. Thus, Set is simple described in terms of the properties of
those objects that belong to the set.
SET is a collection of objects whose properties are well defined so that there is no
question as to whether a certain objects does or does not belong to the set.
The object of set is called “ELEMENTS”. By well defined, we mean that there is a rule
that enables us to determine whether a given object is an element of the set. If a set has no
ELEMENTS, it is called “EMPTY SET or NULL SET”, and is denoted by the symbol Ø.
Sets can be identified or named with the use of capital letters or symbols.
Set can be described by using two methods:
1. ROSTER METHOD - indicate a set by listing the elements, separated by
commas and enclosed them in braces { }.
FOR EXAMPLE: D = { 3, 5, 7} – read as “set of odd numbers whose elements are 3,5,7”
E = { a, b, c, ….., z} – read as “set of English alphabet whose elements starts
from a, b, c up to z”

3. 2. SET – BUILDER NOTATION or RULE METHOD - indicate a set by describing the
elements.
FOR EXAMPLE: D = { x | x = 2 } – read as “ D is the set of all x such that x is 2”
E = { y | y > 2 } – read as “ E is the set of all y such that y is greater than 2”
Set are distinct, we never repeat elements. For Example, { 1, 2, 3, 2 }; the correct listing is { 1, 2, 3 }.
Because a set is a collection, the order in which the elements are listed is IMMATERIAL. It can be
{ 1, 2, 3 } or { 3, 2, 1 } or { 1, 3, 2 } and so on, all represent the same set.
If every element of a set A is also an element of a set B, then we say that A is a subset of B and we
write A Є B. If two A and B have the same elements, then we say that A equals B and write A = B.
FOR EXAMPLE:
A = { 1, 2, 3 } A = { 1, 2, 3 }
B = { 1, 2, 3, 4, 5, 6 } B = { 1, 2, 3 }
1. A Є B = { 1, 2, 3 } Є { 1, 2, 3, 4, 5, 6 } 1. A = B = { 1, 2, 3 } = { 1, 2, 3 }

4.  OPERATIONS ON SET
These are the operations which involves in a sets.
1. If A and B are sets, the UNION of A with B denoted by A U B, is a set consisting of elements that belong to either A or
B.
FOR EXAMPLE:
Let A = { 1, 3, 5, 8 }, B = { 3, 5, 7} and C = { 0, 6, 7, 9}
Give the elements of;
(a) A ∪ B
(b) A ∪ C
Solutions:
(a) A ∪ B = { 1, 3, 5, 8 } ∪ { 3, 5, 7} = { 1, 3, 5, 7, 8 }
(b) A ∪ C = { 1, 3, 5, 8 } ∪ { 0, 6, 7, 9} = { 0, 1, 3, 5, 6, 7, 8, 9 }
2. If A and B are sets, the INTERSECTION of A with B denoted by A ∩ B, is a set consisting of elements that belong to
both A and B.
FOR EXAMPLE:
Let A = { 1, 3, 5, 8 }, B = { 2, 3, 5, 7} and C = { 0, 5, 7, 9}
Give the elements of;
(a) A ∩ B
(b) A ∩ C
Solutions:
(a) A ∩ B = { 1, 3, 5, 8 } ∩ { 2, 3, 5, 7} = { 3, 5 }
(b) B ∩ C = { 2, 3, 5, 7} ∩ { 0, 5, 7, 9} = { 5, 7 }
(b) B ∩ (A ∪ C ) = { 2, 3, 5, 7 } ∩ [{ 1, 3, 5, 8 } ∪ { 0, 5, 7, 9} ] = { 2, 3, 5, 7} ∩ { 0,1, 3, 5, 7, 8, 9 }
= { 3, 5, 7 }

5. 3. Consider A as a set, the COMPLEMENT of A denoted by A’ read as “A prime” refers to the set whose elements are not
in A but elements of the universal set U. Universal set U the set consisting of all the elements that we wish to consider.
FOR EXAMPLE:
We first designate the Universal set U,
Let U = { 1, 2, 3, 4, 5, 6, 7, 8, 9 }
Let A = { 3, 5, 7 }
Let B = { 2, 3, 4, 5 }
Give the elements of;
(a) A’
(b) B’
(c) (A ∪ B )’
(d) (A ∩ B )’
Solutions:
(a) The elements of A are 3, 5, 7. Thus, the elements of A’ = { 1, 2, 4, 6, 8, 9 }
(b) The elements of B are 2, 3, 4, 5. Hence, the elements of B’ = { 1, 6, 7, 8, 9 }
(c) To find the elements of (A ∪ B )’, we determine first the elements of A ∪ B. Hence,
A ∪ B = { 2, 3, 4, 5, 7 }
So the elements of (A ∪ B )’ = { 1, 6, 8, 9 }
(d) To find the elements of (A ∩ B )’, we determine first the elements of A ∩ B. Hence,
A ∩ B = { 3, 5 }
So the elements of (A ∩ B )’ = { 1, 2, 4, 6, 7, 8, 9 }

6. It is often helpful to draw pictures of sets. Such pictures, called Venn diagrams, represent sets as circles enclosed in a
rectangle, which represents the universal set. Such diagrams often help us to visualize various relationships among sets.
See Figure 1.
If we know that A ∈ 𝐵, we might use the Venn diagram in Figure 2(a). If we know that A and B have no elements in
common, that is, if A ∩ 𝐵 = ∅ we might use the Venn diagram in Figure 2(b). The sets A and B in Figure 2(b) are said to
be disjoint.
Figures 3(a), 3(b), and 3(c) use Venn diagrams to illustrate the definitions of intersection, union, and complement,
respectively.

7. Lets have some example:
Use the information supplied in the Figure.
1. How many are in A?
2. How many are in A or B?
3. How many are in A and C?
4. How many are not in B?
5. How many are in neither A nor C?
6. How many are in B but not in C?
SOLUTION:
1. Tignan lamang ung mga numero na nasa loob ng Circle A, A = 20 + 2 + 1 + 6 = 29
2. Tignan kung anu-ano ang mga nasa loob ng circle A at B, A ∪ B = 0 + 1 + 2 + 5 + 6 + 20 = 34
3. Ang iconsider lang ay yung numero na parehong nasa loob ng circle A at C, A ∩ C = 1 + 6 = 7
4. Ang iconsider ay yung mga numerong hindi naka pa loob sa circle B, B’ = 20 + 1 + 4 + 20 = 45
5. Ang iconsider ay yung mga numerong hindi naka pa loob sa A at C, (A ∪ C)’ = 20 + 5 = 25
6. Tignan ang circle B at C, then iconsider lang yung hindi naka paloob sa C, B = 5 + 2 = 7

8. 1.2 REAL NUMBERS
Numbers are essentials not only in arithmetic and algebra but also to other fields like economics
and business. The developments of science and technology needs a scientific number system. The most
important number system in algebra is the real numbers. It is the result of gradual development in the different
number systems.
 NATURAL NUMBERS – are the counting numbers. N = { x | x = 1, 2, 3, …. }
 WHOLE NUMBERS – are the counting numbers together with the zero (0). W = { x | x = 0, 1, 2, 3, …. }
 INTEGERS – are the negative and positive numbers together with the zero (0). I = { x | x = …, -3, -2, -1, 0, 1, 2, 3, …. }
 RATIONAL NUMBERS – are the numbers that can be expressed as a quotient
𝑎
𝑏
of two integers. The integer a is
called numerator, and the integer b, which cannot be 0 (b ≠ 0), is called denominator. These numbers can either
be repeating or terminating when it is presented in decimals.
Repeating Numbers – are usually presented with a dash sign written above the number or numbers being
repeated.
For Example:
1.
3
11
= 0.272727… 2.
2
3
= 0.66666…
= 0.27 = 0.6
Terminating Numbers – usually defined as a decimal number that contains a finite number of digits
after the decimal point .
For Example:
1.
1
4
= 0.25 2.
5
2
= 2.5
 IRRATIONAL NUMBERS – numbers that may represented by a decimal but it is neither repeats nor terminates. In
other words, irrational numbers cannot be written in the form
𝑎
𝑏
where a, b are integers and b ≠ 0. For Example,
2 , 3 , and 5.
 REAL NUMBERS – is the union of the set of rational numbers with the set of irrational numbers.

9.  APPROXIMATIONS
Every decimal may be represented by a real number (either rational or irrational), and every real
number may be represented by a decimal. In practice, the decimal representation of an irrational number is
given as an approximation. For example, using the symbol ≈ (read as “approximately equal to”), we can write
2 ≈ 1.4142 𝜋 ≈ 3.1416
In approximating decimals, we either round off or truncate to a given number of decimal places.
The number of places establishes the location of the final digit in the decimal approximation.
Truncation: Drop all the digits that follow the specified final digit in the decimal.
Rounding: Identify the specified final digit in the decimal. If the next digit is 5 or more, add 1 to the final digit; if
the next digit is 4 or less, leave the final digit as it is. Then truncate following the final digit.
FOR EXAMPLE: Approximate 20.98752 to two decimal places by
(a) Truncating
(b) Rounding
Solution:
For 20.98752, the final digit is 8, since it is two decimal places from the decimal point.
(a) To truncate, we remove all digits following the final digit 8. The truncation of 20.98752 to two decimal
places is 20.98.
(b) The digit following the final digit 8 is the digit 7. Since 7 is 5 or more, we add 1 to the final digit 8 and
truncate. The rounded form of 20.98752 to two decimal places is 20.99.

10.  OPERATIONS
In algebra, we use letters such as x, y, a, b, and c to represent numbers. The symbols used in algebra
for the operations of addition, subtraction, multiplication, and division are +, −, ∙ and /. The words used to
describe the results of these operations are sum, difference, product, and quotient. Table 1 summarizes these
ideas.
TABLE 1.1
In algebra, we generally avoid using the multiplication sign and the division sign so familiar in
arithmetic. Notice also that when two expressions are placed next to each other without an operation symbol,
as in or in parentheses, as in it is understood that the expressions, called factors, are to be multiplied.
We also prefer not to use mixed numbers in algebra. When mixed numbers are used, addition is
understood; for example, 2
3
4
means 2 +
3
4
. In algebra, use of a mixed number may be confusing because the
absence of an operation symbol between two terms is generally taken to mean multiplication. The expression
2
3
4
is therefore written instead as 2.75 or as
11
4
.
The symbol “=“, called an equal sign and read as “equals” or “is,” is used to express the idea that the
number or expression on the left of the equal sign is equivalent to the number or expression on the right.
OPERATION SYMBOL WORDS
Addition a + b Sum: a plus b
Subtraction a – b Difference: a minus b
Multiplication a∙b, (a) ∙b, a∙(b), (a) ∙(b)
ab, (a)b, a(b), (a)(b)
Product: a times b
Division a/b or
𝑎
𝑏
Quotient: a divided by b

11. Lets have some example, writing statements using symbols.
(a) The sum of 2 and 7 equals 9. In symbols, this statement is written as 2 + 7 = 9
(b) The product of 3 and 5 is 15. In symbols, this statement is written as 3 ∙ 5 = 15
 EVALUATE NUMERICAL EXPRESSIONS
Rules for the Order of Operations
1. Begin with the innermost parentheses and work outward. Remember that in dividing two
expressions the numerator and denominator are treated as if they were enclosed in parentheses.
2. Perform multiplications and divisions, working from left to right.
3. Perform additions and subtractions, working from left to right.
FOR EXAMPLE:
Evaluate each expression.
a) 8 ∙ 2 + 3 b) 5 ∙ (3 + 4) + 2 c) 2 + [4 + 2 ∙ (10 + 6)]
SOLUTION:
a) 8 ∙ 2 + 3 = 16 + 3 = 19
b) 5 ∙ (3 + 4) + 2 = 5 ∙ 7 + 2 = 35 + 2 = 37
c) 2 + [4 + 2 ∙ (10 + 6)] = 2 + (4 + 2 ∙ 16) = 2 + (4 + 32) = 2 + 36 = 38

12. 1.3 PROPERTIES OF REAL NUMBERS
1. COMMUTATIVE PROPERTY
a) a + b = b + a (commutative property for addition)
b) ab = ba (commutative property for multiplication)
This statement states that the sum or product of two or more numbers is not affected regardless of the orders to which the values are added or multiplied.
FOR EXAMPLE:
3 + 7 = 7 + 3 = 10
7∙2 = 2∙7 = 14
2. ASSOCIATIVE PROPERTY
a) (a + b) + c = a + (b + c) (associative property for addition)
b) (ab)c = a(bc) (associative property for multiplication)
This statement above says that the sum or product of three numbers is not affected regardless of the grouping of values.
FOR EXAMPLE:
(4 + 8) + 2 = 4 + (8 + 2) = 14
(7∙2) ∙3 = 7∙ (2 ∙3) = 14
3. DISTRIBUTIVE PROPERTY
a) a(b + c) = ab + ac
b) (b + c) a = ab + ac
This statement above says that the sum of two or more numbers when multiplied to another number is equal to multiplying each separately then adding the products.
FOR EXAMPLE:
2(3 + 6) = 2(3) + 2(6)
2(9) = 6 + 12
18 = 18
4. IDENTITY PROPERTY
a) a + 0) = a
b) a ∙ 1 = a
The real number 0 is known as the additive identity and the number 1 is called multiplicative identity Statements a, explains that any number added to 0 will result to the
number statement b. This denotes that if a number is multiplied to 1, than the result is that number.
FOR EXAMPLE:
5 + 0 = 5
5 ∙ 1 = 5
5. INVERSE PROPERTY
The additive inverse property states that adding a number and its inverse results in a sum of 0. The multiplicative inverse property states that multiplying a nonzero
number with its inverse results in a product of 1.

13. a) ADDITIVE INVERSE PROPERTY
a – b = a + (-b)
NOTE: Don’t assume that – a is a negative number. Whether – a is a negative or positive depends on the value of a.
TABLE 1.2 PROPERTIES OF NEGATIVE
PROPERTIES OF NEGATIVE
PROPERTY EXAMPLE
1. (– 1)a = – a (– 1)5 = – 5
2. – (– a) = a – (– 5) = 5
3. (– a)b = – ab (– 5)2 = – (5)(2)
4. (– a)(– b) = ab (– 5)(– 2) = (5)(2)
5. – (a + b) = – a – b – (5 + 2) = – 5 – 2
6. – (a – b) = – a + b – (a – b) = – 5 + 2

14. b) MULTIPLICATIVE INVERSE PROPERTY
a/b = a ∙
𝟏
𝒃
We write a ∙
𝟏
𝒃
as simply
𝒂
𝒃
. We refer to
𝒂
𝒃
as the 𝐪𝐮𝐨𝐭𝐢𝐞𝐧𝐭 of a and b or as the fraction a over b; a is the numerator and b is the
denominator or (divisor).
TABLE 1.3 PROERTIES OF FRACTION
PROPERTIES OF FRACTION
PROPERTY EXAMPLE DESCRIPTION
1.
𝒂
𝒃
∙
𝒄
𝒅
=
𝒂𝒄
𝒃𝒅
𝟐
𝟑
∙
𝟓
𝟕
=
𝟐∙𝟓
𝟑∙𝟕
=
𝟏𝟎
𝟐𝟏
When multiplying fractions, multiply
numerators and denominators.
2.
𝒂
𝒃
÷
𝒄
𝒅
=
𝒂
𝒃
∙
𝒅
𝒄
𝟐
𝟑
÷
𝟓
𝟕
=
𝟐
𝟑
∙
𝟕
𝟓
=
𝟏𝟒
𝟏𝟓
When dividing fractions, get the reciprocal of
the divisor and then multiply.
3.
𝒂
𝒄
+
𝒃
𝒄
=
𝒂+𝒃
𝒄
𝟐
𝟓
+
𝟑
𝟓
=
𝟐+𝟕
𝟓
=
𝟗
𝟓
When adding fractions with the same
denominator, add the numerators.
4.
𝒂
𝒃
+
𝒄
𝒅
=
𝒂𝒅 + 𝒃𝒄
𝒃𝒅
𝟐
𝟓
+
𝟑
𝟕
=
𝟐(𝟕) +𝟓(𝟑)
𝟑𝟓
=
𝟐𝟗
𝟑𝟓
When adding fractions with different
denominators, find the common denominator.
Then add the numerators.
5.
𝒂𝒄
𝒃𝒅
=
𝒂
𝒃
𝟐∙𝟓
𝟑∙𝟐
=
𝟓
𝟑
Cancel numbers that are common factors in the
numerator and denominator.
6.
𝒂
𝒃
=
𝒄
𝒅
, 𝑡ℎ𝑒𝑛 𝒂𝒅 = 𝒃𝒄 𝟐
𝟑
=
𝟔
𝟗
, 𝑡ℎ𝑒𝑛 𝟐(𝟗) = 𝟑(𝟔) Cross - multiply

15. 1.4 THE REAL NUMBER LINE AND ORDER
The real numbers can be represented by points on a line, called real number line. We choose an
arbitrary reference point O, called the Origin, which corresponds to the real number 0.
Figure 1.1 Real Number Line
The point 1 unit to the right of O corresponds to the number 1. The distance between 0 and 1
determines the scale of the number line. For example, the point associated with the number 2 is twice as
far from O as 1. Notice that an arrowhead on the right end of the line indicates the direction in which the
numbers increase. Points to the left of the origin correspond to the real numbers – 1, – 2, and so on.
Figure 1.1 also shows the points associated with the rational numbers –
1
4
and
1
4
with the irrational
numbers 2 and 𝜋.
The real number associated with a point P is called the coordinate of P, and the line whose points have
been assigned coordinates is called the real number line.
POSITIVE DIRECTION

16.  The real number line consists of three classes of real numbers, as shown in Figure 1.2
Figure 1.2
1. The negative real numbers are the coordinates of points to the left of the origin O.
2. The real number zero is the coordinate of the origin O.
3. The positive real numbers are the coordinates of points to the right of the origin O.
Multiplication Properties of Positive and Negative Numbers
1. The product of two positive numbers is a positive number.
(+) ∙ (+) = +
2. The product of two negative numbers is a positive number.
(–) ∙ (–) = +
3. The product of a positive number and a negative number is a negative number.
(+) ∙ (–) = – and (–) ∙ (+) = –

17. An inequality is a statement in which two expressions are related by an inequality symbol.
The expressions are referred to as the sides of the inequality. Inequalities of the form a < b
or a > b are called strict inequalities, whereas inequalities of the form 𝑎 ≤ 𝑏 or 𝑎 ≥ 𝑏 are
called non-strict inequalities. Based on the discussion so far, we conclude that:
FOR EXAMPLE
(a) On the real number line, graph all numbers x for which 4 > x.
(b) On the real number line, graph all numbers x for which x ≤ 5.
SOLUTION:
(a) See Figure 1.4 Notice that we use a left parenthesis to indicate that the number 4 is not part of the
graph.
(a) See Figure 1.5 Notice that we use a right bracket to indicate that the number 5 is part of the graph.
a < 0 is equivalent to a is positive
a > 0 is equivalent to a is negative
FIGURE 1.4
FIGURE 1.5

18.  GRAPH OF INEQUALITIES
An important property of the real number line follows from the fact that,
given two numbers (points) a and b, either a is to the left of b, or a is
at the same location as b, or a is to the right of b. See Figure 1.3
If a is to the left of b, we say that “a is less than b” and write a < b If a
is to the right of b, we say that “a is greater than b” and write a > b If a
is at the same location as b, then a = b If a is either less than or equal
to b, we write 𝑎 ≤ 𝑏 Similarly, 𝑎 ≥ 𝑏 means that a is either greater than or
equal to b. Collectively, the symbols and are called inequality
symbols. Note that and mean the same thing. It does not matter
whether we write or Furthermore, if or if then the difference is positive.
FOR EXAMPLE:
1. 2 < 6
2. 9 > 5
3. – 3 > – 10
4. 4 < 10
5. 4 > – 1
Note: That the inequality symbol
always points in the direction of the
smaller number.
FIGURE 1.3

19.  SETS AND INTERVALS
Certain sets of real numbers, called intervals, occur frequently in calculus an d correspond geometrically
to line segments. For example, if a < b, then the open interval from a to b consists of all numbers
between a and b is denoted by the symbol (a, b). Using the set builder notation, we can write
Note that the endpoints a and b, are excluded from this interval. This fact is indicated by parentheses ( )
in the interval notation and the open circles on the graph of interval in Figure 1.6.
The closed interval from a to b is the set
Here the endpoints of the interval are included. This indicated by the square brackets [ ] in the interval
notation and solid circles on the graph of the interval in Figure 1.7.
(a, b) = { x | a < x < b }
FIGURE 1.6
[a, b] = { x | a < x < b }
FIGURE 1.7

20. TABLE 1.4 NINE POSSIBLE TYPES OF INETRVALS
NOTATION SET DESCRIPTION GRAPH
( a, b ) { x| a < x < b }
[ a, b ] { x| a ≤ x ≤ b }
[ a, b ) { x| a ≤ x < b }
( a, b ] { x| a < x ≤ b }
( a, ∞ ) { x| a < x }
[ a, ∞ ) { x| a ≤ x }
(–∞, b ) { x| x < b }
(– ∞, b ] { x| x ≤ b }
(– ∞, ∞ ) R (set of all real number)

21. Lets have some example:
Express each interval in terms of inequalities, and the graph the interval.
1. [-1, 2) = { x | -1 ≤ x < 2 }
2. [1.5, 4] = { x | 1.5 ≤ x ≤ 4 }
3. (-3, ∞) = { x | -3 < x }
Graph each set.
1. (1, 3) ∩ [2, 7]
Let us first graph the given intervals
(1, 3) = { x | 1 < x < 3 }
[2, 7] = { x | 2 ≤ x ≤ 7 }
The intersection of two intervals consists of the numbers that are in both intervals.
(1, 3) ∩ [2, 7] = { x | 1 ≤ x < 2 and | 2 ≤ x ≤ 7 }
= { x | 2 ≤ x < 3 } = [2, 3)

22.  ABSOLUTE VALUE AND DISTANCE
The absolute value of a number a is the distance from 0 to a on the number line. For example, is 4 units
from 0, and 3 is 3 units from 0. See Figure 1.8. Thus, the absolute value of - 4 is 4, and the absolute value
of 3 is 3.
TABLE 1.5 PROPERTIES OF ABSOLUTE VALUE
FIGURE 1.8
Definition of Absolute Value
If a is a real number, then the absolute value of a is denoted by |a| and
defined by
|a| = a if a ≥ 0 and |a| = – a if a < 0
PROPERTY EXAMPLE DESCRIPTION
1. |a| ≥ 0 |– 3| = 3 ≥ 0 The absolute value of a number is always positive or zero
2. |a| = |– a| |5| = |– 5| A number and its negative have the same absolute value
3. |ab| = |a| |b| |– 2∙5| = |– 2| |5| The absolute value of a product is the product of the
absolute value
4. |
𝒂
𝒃
| =
|𝒂|
|𝒃|
|
5
– 2
| =
|5|
|– 2|
The absolute value of a quotient is the quotient of the
absolute values.

23. Distance Between Points on the Real Line
If a is a real number, then the distance between the points a and b
on the real line is
d(a, b) = |b – a|
FOR EXAMPLE:
Let P, Q, and R be points on a real number line with coordinates – 5, 7 and – 3 respectively. Find the
distance;
(a) between P and Q
(b) between Q and R
SOLUTION:
(a) d(P, Q) = |7 – (– 5)| = |12| = 12
(b) d(P, Q) = |– 3 – 7| = | – 10| = 10

24. 1.5 INTEGER EXPONENT
A product of identical numbers is usually written in exponential notation.
FOR EXAMPLE:
1. (–3)4
= –3 ∙ –3 ∙ –3 ∙ –3 = 81
2. –34
= – (3 ∙ 3 ∙ 3 ∙ 3) = – 81
FOR EXAMPLE:
1. (
4
7
)0
= 1
2. 𝑥−1
=
1
𝑥1 =
1
𝑥
3. (–2)–3
=
1
(–2)3 =
1
−8
= −
1
8
Exponential Notation
If a is any real number and n is a positive integer, then the nth power of a
𝑎𝑛
= 𝑎 ∙ a ∙ … ∙ a
The number a is called the base and n is called exponent.
Note: The distinction between
(–3)4
and –34
. 𝐼𝑛(–3)4
the exponent
applies to –3, but in –34
the exponent
applies only to 3.
Zero and Negative Exponents
If a ≠ 0 is any real number and n is a positive integer, then
𝑎0
= 1 𝑎𝑛𝑑 𝑎−𝑛
=
1
𝑎−𝑛

25. TABLE 1.9 LAWS OF EXPONENTS
LAWS EXAMPLE DESCRIPTION
1. 𝑎𝑚
𝑎𝑛
= 𝑎𝑚+𝑛
32 ∙ 33= 32+3 = 35 To multiply two powers of the same number,
add the exponent
2.
𝑎𝑚
𝑎𝑛 = 𝑎𝑚−𝑛 35
32 = 35−2
= 33 To divide two powers of the same number,
subtract the exponent
3. (𝑎𝑚
)𝑛
= 𝑎𝑚𝑛
(32
)5
= 310 To raise a power to a new power, multiply the
exponents
4. (𝑎𝑏)𝑛
= 𝑎𝑛
𝑏𝑛
(3 ∙ 9)4
= 34
∙ 94 To raise a product to a power, raise each
factor to the power
5. (
𝑎
𝑏
)𝑛=
𝑎𝑛
𝑏𝑛 (
3
4
)2
=
32
42
To raise a quotient to a power, raise both
numerator and denominator to the power
6. (
𝑎
𝑏
)−𝑛=
𝑏𝑛
𝑎𝑛 (
3
4
)−2=
42
32
To raise a fraction to a negative power,
invert/reverse the fraction and change the
sign of the exponent
7.
𝑎−𝑚
𝑏−𝑛 =
𝑏𝑛
𝑎𝑚
3−2
4−5 =
45
32
To move a number raised to a power from
numerator to denominator or from
denominator to numerator, change the sign
of the exponent

26. Lets have some examples:
Using the laws of exponent
1. 𝑥4𝑥7 = 𝑥4+7 = 𝑥11
2. 𝑦4
𝑦−7
= 𝑦4−7
= 𝑦−3
=
1
𝑦3
3.
𝑐9
𝑐4 = 𝑐9−4 = 𝑐5
4. (𝑠4)5= 𝑠4(5) = 𝑠20
5. (3𝑥4)3= 33𝑥4(3) = 27𝑥12
6. (
𝑥
2
)5=
𝑥5
25 =
𝑥5
32
Lets have some examples:
Simplifying expressions with exponent
1. (2𝑎3𝑏2) 3𝑎𝑏4)3 = (2𝑎3𝑏2 33𝑎3(𝑏4)3
= (2𝑎3
𝑏2
) 27𝑎3
𝑏12
= (2)(27) 𝑎3
𝑎3
𝑏2
𝑏12
= 54 𝑎6𝑏14
2. (
𝑥
𝑦
)3(
𝑦2𝑥
𝑧
)4 =
𝑥3
𝑦3
𝑦2 4
𝑥4
𝑧4
=
𝑥3
𝑦3
(
(𝑦8)𝑥4
𝑧4
)
= 𝑥3
𝑥4
𝑦8
𝑦3
1
𝑧4
=
(𝑥7)𝑦5
𝑧4

27. Lets have some examples:
Simplifying expressions with negative
exponent
1.
(6𝑠𝑡−4)
2𝑠−2𝑡2 =
(6𝑠𝑠2)
2𝑡2𝑡4
=
3𝑠3
𝑡6
2. (
𝑦
3𝑧3)−2
= (
3𝑧3
𝑦
)2
=
32 𝑧3 2
𝑦2 = (
9𝑧6
𝑦2 )
 SCIENTIFIC NOTATION
Measurements of physical quantities can range from very small to very large. For example, The
mass of a proton is approximately 0.00000000000000000000000000167 kilogram and the mass of Earth is
about 5,980,000,000,000,000,000,000,000 kilograms. These numbers obviously are tedious to write down
and difficult to read, so we use exponents to rewrite each.
Converting a Decimal to Scientific Notation
To change a positive number into scientific notation:
1. Count the number N of places that the decimal point must be moved to arrive at a number x where 1
≤ x < 10.
2. If the original number is greater than or equal to 1, the scientific notation is x 10−𝑁
. If the original
number is between 0 and 1, the scientific notation x 10−𝑁
.
When a number has been written as the product of a number x,
where 1 ≤ x < 10 times a power of 10, it is said to be written in
scientific notation.

28. Lets have some example:
1. 4 x 1013
= 40, 000, 000, 000, 000
2. 1.66 x 10−24
= 0.00000000000000000000000166
3. 56, 920 = 5.692 x 104
4. 0.000093 = 9.3 x 10−5

29. THANK YOU!!