1. Chapter 1 Review Topic in Algebra 1
1. Set of real numbers
1.1 Real number is a set of rational numbers and the set of irrational numbers make up.
If the numbers are repeating or terminating decimal they called rational number. The square roots of
perfect squares also name rational number.
For examples:
1) √0.16
2) 0.666
3)
1
3
4)
10
9
5)
9
6
If the numbers are not repeating or terminating decimals. They called irrational number.
For examples:
1) π
2) √2
3) 0.61351
4) √8
5) √11
Exercise 1.1
Direction: Determine whether each statement is true or false.
1. Every integer is also a real number.
2. Every irrational number is also an irrational number.
3. Every natural number is also a whole number.
4. Every real number is also a rational number.
State whether each decimal represents a rational o irrational number.
5. √4
6. √5
7. 0
8. 3
9. 0.63586358
10. √866
1.1.1 Properties of real numbers
Let us denote the set of real numbers by 푅. These properties are statement derived from the basic
axioms of the real numbers system. Axioms are assumptions on operation with numbers.
Axioms of Equality
Let a, b, c, d ∈ R
1. Reflexive Law
If a=a
2. Symmetric Law
If b=c then c=b
3. Transitive Law
If b=c and c=d then b=d
4. Additional Law of Equality
2. If a=b then a+c=b+c
5. Multiplication Law of Equality
If a=b then a.c=b.c
Axioms for Addition and Multiplication
Let a, b, c, d, ∈ R
1) A. Closure property for addition
a+b ∈ R
Examples:
1) 3+3=6
2) 7+(-4)=3
3) -8+4=-4
B. Closure property for multiplication
a.b ∈ R
Examples:
1) 3(7)=21
2) -8(3)=-24
3) 0.11=0
2) A. Commutative prroperty for addition
a+b=b+a
Examples:
1)
1
2
+ 7 = 7 +
1
2
2) 0.3 + (−
5
6
) = −
5
6
+ 0.3
3)
1
3
+ 21 = 21 +
1
3
B. Commutative prroperty for multiplication
a.b=b.a
Examples:
1)
4
5
4
5
(22) = 22 (
)
2) 6.3=3.6
3)
10
9
10
9
(−25) = −25 (
)
3) A. Associative property for addition
(a+b)+c=a+(b+c)
Examples:
1) (3+7)+0.4=3+(7+0.4)
2) (0.36+89)+
1
2
= 0.36 + (89 +
1
2
)
3
5
3) (
+ 0.8) +
3
8
=
3
5
+ (0.8 +
3
8
)
B. Associative property for multiplication
(a.b).c=a.(b.c)
Examples:
1) (3.x).y=3.(x.y)
2) [5(7)] 1
4
1
4
= 5 [7 (
)]
3) [3푥(6푥)]]5 = 3푥[6푥(5)]
4) Identity property for multiplication
3. a.1=a
Examples:
1) 1.a3=a3
2)
3
7
(1) =
3
7
3) 3.1=3
5) A. Inverse property for addition
a+(-a)=0
Examples:
1) 6+(-6)=0
2) 10+(-10)=0
3) -3+3=0
B. Inverse property for multiplication
푎.
1
푎
= 1
Examples:
1) -2(−
1
2
)=1
2) 8(
1
8
)=1
3) -6(-
1
6
)=1
6) Distributive property of multiplication over addition
a(b+c)=ab+ac
Examples:
1) 3(4+6)=3(4)+3(6)
2) -6(7+1)=-6(7)+[-6(1)]
3) a(7+5)=7a+5
Exercise 1.1.1
Determine which real number property is shown by each of the following.
1. −
1
4
+
1
4
= 0
2. 2(1)=2
3.
1
4
(4)=1
4. -7+(-4)=-4+(-7)
5. 0.3(0)=0.3
6. 5[3+(-1)]=5(3-1)
7. (8+
9
8
)+0.45=8+(
9
8
+0.45)
8. 5(8+8)=5(8)+5(8)
9. 6x+(8x+10)=(6x+8x)+10
10. 5a+2b=2b+5a
4. 1.2 Exponents and Radicals
In the expression 훼푛, α is the base and 혯 is the exponent. The expression 훼푛 means that the
value α is multiplied 혯 times by itself.
Examples:
1) 63= 6.6.6
=216
2) 56= 5.5.5.5.5
=15625
3) 42= 4.4
=16
1.2.1 Integral and zero exponents
Laws of Integral and Zero Exponents
Theorem 1:
For any real number α, (α≠ 0)
푎0 = 1
Examples:
1) (6푎0 + 3)0=1
2) 6α0+70=6(1)+1=7
3) 2α0+70=2(1)+1=3
Theorem 2:
For any real numbers α,
αm. α혯= αm+n
where m and n are integers.
Examples:
1) α5.α4=푎5+4 = 푎9
2) 4푥푦2(2푥2푦2) = 8푥1+2푦2+2 = 8푥3푦4
3) 푥푎+3. 푥푎+4 = 푥2푎+7
Theorem 3:
For any real numbers a+b,
(ab)n=anbn,
where n is any integer.
Examples:
1) (5x)2=55x2=25x2
2) (-2x)3=-23x3=-8x3
3) [x(x-3)]2=x2(x-3)2
=x2(x2-6x+9)
=x4-6x3+9x2
Theorem 4:
For any real numbers a
(am)n=amn
where m and n are integers.
Examples:
1) (-x2)3=-x2(3)=-x6
2) [(3x+4)2]3=(3x+4)6
3) (-x2y3z)4=-x8y12z4
Theorem 5:
5. For any real numbers a and b (b≠0),
푎
푏
(
)푛 =
푎푛
푏푛
where n is any integer.
Examples:
1) (
푎2
푏3)2 =
푎4
푏6
2) (
3
4
)3 =
33
43 =
27
64
3) (
푥
푦+2
)2=
푥2
(푦+2)2 =
푥2
푦2+4푦+4
Theorem 6:
For any real numbers a(a≠0),
푎푚
푎푛 = 푎푚−푛
where m and n are integers.
Examples:
1)
푎7
푎5 = 푎7−5=푎2
2)
푥3푦4푧5
푥푦푧
= 푥3−1푦4−1푧5−1 = 푥2푦3푧4
3)
푥4푦4
푥4푦4 = 푥4−4푦4−4 = 푥0푦0 = 1(1) = 1
Theorem 7:
For any real numbers a(a≠0),
푎−푛 =
1
푎푛
Where n is any positive integer.
Examples:
1) 3푥3푦−2=
3푥3
푦2
2) (4푥2푦)−2 =
1
(4푥2푦)2 =
1
8푥4푦2
3) (푥2 + 푦)−2 =
1
(푥2+푦)2 =
1
푥4+푦2
Exercises 1.2.1
Simplify and express the following expressions with positive and negative integrals only.
1. 50
2.
10푚4
30m
3.
16푏4푐
−4푏푐3
4. 푦3. 푦4
5. (5푥푦) 6
6. (푎푏) 3
7. (푥3푦2)3
6. 8. [(−5)2]2
9.
푥5푦6
푥푦
=
10.
푎7
푎3
1.2.2 Fractional Exponents: Radicals
Since not all numbers are integers, we can’t expect exponents to always whole number or
zero. Exponents can be form fractional. Fractional exponents may seem unfamilliar for they are
usually expressed as radicals.
For expression 푥
1
2 is the same as √2 (read as square root of 2), and 푥
2
3 is the same as
3√푥2 (read as cube root of x squared). The expression 푛√푎푚 is called a radical. The symbol √ is
called a radical sign, where n is the index, a is the radicand and m is the power of the radicand.
푎
푚
푛
=푛√푎푚
Laws of Radicals
Theorem 1:
For any real numbers a,
√푎푛 = 푎 푛
Examples:
1) √42 = 4
2) 3√(푥2푦)3=푥2푦
3) 3√33=3
Theorem 2:
For any real numbers a,and b.
√푎 푛 . √푏 푛 =√푎푏 푛
Examples:
1) √3. √3 = √3.3 = √9=3
2) √4. √3 = √4.3 = √12
3) √푎. √푏 = √푎. 푏
Theorem 3:
For any real numbers a,and b, (b≠0)
√푎 푛
√푏 푛 = √
푎
푏
푛
Examples:
1)
3√푎
3√푏 = √
푎
푏
3
2)
√4
√5
4
5
= √
3)
4√푥
4√푦 = √
푥
푦
4
Theorem 4:
For any real numbers a ,
7. √푎 푚푛 = √ √푎 푚 푛
= √ √푎 푛 푚
Examples:
1) √64 6 = √√64 3 = √8 3 = 2
2) √16 4 = √√16 2 = √4 2 =2
3) √100 3 = √100 3 =√100 = 10
Theorem 5:
For any real numbers a
k푛√푎푘푚 = 푛√푎푚
Examples:
1) 6√24=2.3√22.2 = 3√22 = √4 3
2) 6√93 = 3.2√93.1 = √9 2 =3
3) √ 0.16
1.2.1 Addition and Sutraction of Radicals
To add and subtract radicals, first we need to combine the like terms with similar
radicals.
Examples:
1) √2 + 3√2 − 2√2 = 2√2
2) √8 + √18 + √32 = √4.2 + √9.2+√16.2 = 2√2 + 3√2 + 4√2 = 9√2
3) 푦√푥3푦 − √푥3푦3 + 푥√푥푦3 = 푦√푥2. 푥푦 − √푥2. 푥. 푦2. 푦 + 푥√푥. 푦2. 푦 = 푥푦√푥푦 −
푥푦√푥푦 + 푥푦√푥푦 = 푥푦√푥푦
1.2.2 Multiplication and Division of Radicals
To multiply and divide radicals with the same index, multiply, or divide the radicals and
copy the common index.
Examples:
1) √3.√3 = √32 = 3
2) 3√푥푦.√푥2푦 3 . √푥푧 3 =√푥푦. 푥2푦. 푥푧 3 = √푥4푦2푧 3 = 푥 √푥푦2푧 3
3) √16 3 ÷√−2 3 =3√16 ÷(−2)=√−8 3 = −2
Exercise: 1.2.2
Simplify and solve.
1. (5√2)(3√6)
2. (3푎3√4푥2)(43√3푥푦)
3. 4√
9
16
4. √2(3+√3)
5. 5√2+3√2
6. √18 − 2√27 + 3√3 − 6√8
7. √16푏 + √4푏
9. 1.3 polynomials
Polynomials was used to describe any algebraic expression. The algebraic
expression, 5x+4 and x3+x2+1 are examples of polynomials in variable x. A polynomial
with just one term 2x is called a monomial. If the polynomial is the sum or difference of
two terms as in -9x+7, then it is called a binomial. If it has three terms like x2+2x+1, then
it is called a trinomial. In general a polynomial consisting of a sum of any numbers of
terms is called a multinomial.
In the binomial, 5x+4 the number 5 is called the numerical coefficient of x while
x is the literal coefficient and the numbers 4 is the constant term.
1.3.1 Addition and Sutraction of Polynomials
To determined the sums and differences of polynomials, only the coefficients are combined. By
similar terms are refer to the terms with the same coefficients. Those with different literal coefficient
are called dissimilar or unlike terms.
Examples:
1) Find the sum of 2x-3y+5 and x+2y-1,
=(2x-3y+5)+( x+2y-1)
=2x+x-3y+2y+5-1
=3x-y++4
2) Find the differences between 2x-3y+5 and x+2y-1
=(2x-3y+5)-( x+2y-1)
=2x-3y+5+(-x-2y+1)
=2x-x-3y-2y+5+1
=x-5y+6
3) Subtract 2(4x+2y+3) from 5(2x-3y+1)
=5(2x-3y+1)- 2(4x+2y+3)
=10x-15y+5-8x+4y+6
=2x-11y+11
1.3.2 Multiplication of Polynomials
Examples:
1) 푥푚.푥푛 = 푥푚+푛
2) 푥−2.푥2=푥0 = 1
3) Multiply a+2b+3c by 5m.
= a+2b+3c(5m) in multiplication, we apply the
=5am+10bm+15cm distributive property
1.3.3 Division of Polynomials
To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.
12. 1.5 Rational Expressions
A fraction where the numerator and denominator are polynomials, and is defined for all values
of the variable that do not make the denominator zero.
1.5.1 Reducing Rational Expression to Lowest Terms
We need to lowest term the fraction, if the numerator and denominator have no common
factor.
Examples:
1)
4푎2푏푐3
6푎푏3푐4 =
2.2.푎.푎.푏.푐.푐.푐
2.3.푎.푏.푏.푏.푐.푐.푐.푐
=
2푎
3푏2푐
2)
푥2+2푥푦+푦2
푥2−푦2 =
(푥+푦)(푥+푦)
(푥+푦)−(푥−푦)
푥+푦
푥−푦
=
3)
푥3+8푦3
4푥+8푦
=
푥+2푦(푥2−2푥푦+4푦2
4(푥+2푦)
푥2−2푥푦+4푦2
=
4
1.5.2 Multiplying and Dividing Rational Expressions
In multiplication if
푝
푞
푎푛푑
푟
푠
are rational expressions and q and s are real numbers not equal to 0,
then
푝
푞
.
푟
푠
=
푝푟
푞푠
.
Examples:
1)
4
3
.
1
5
=
4
15
2)
푐
. (푎 + 2푏)(푎 − 푏)
푎2−푏2 푐
=
(푎 + 푏)(푎 − 푏)
. (푎 + 2푏)(푎 − 푏)
푐(푎+2푏)
푎 +푏
=
In dividing algebraic fractions, multiply the dividend by the reciprocal of the divisor. The
reciprocal of a fraction is its multiplicative inverse.
Examples:
1)
4
3
6
5
÷
=
4
3
.
5
6
=
20
18
표푟
10
9
2)
8
7
÷ 3 =
8
7
.
1
2
=
8
14
표푟
4
7
3)
푦2−16
푦−5
÷
2푦 −8
푥푦−5푥
(푦−4)(푦+4)
=
푦 −5
.
푥(푦−5)
2(푦−4)
=
푥푦+4푥
2
13. 1.5.3 Adding and Subtracting Rational Expressions.
To add and subtract rational expressions, it is the important that the least common
denominator is accurately determined.
Examples:
1)
5
6
−
2
3
+
1
8
=
20−16+3
24
=
7
24
2)
4
5
+
3
5
+
2
5
=
4+3+2
5
=
9
5
3) 3푥 − 2푦 +
2푥2−푦2
푥+푦
3푥(푥+푦)−2푦(푥+푦)+2푥2−푦2
=
푥+푦
=
3푥2+3푥푦−2푥푦+2푦2+2푥2−푦2
푥+푦
5푥2+푥푦−3푦2
=
푥+푦
1.5.4 Simplifying Complex Rational Expressions
A factor which contains one or more fractions either in the numerator or denominator or in
both.
Examples:
1)
4
31
3
=
4
3
.
3
1
=
12
3
표푟 4
2)
3
2+1
3
=
3
6+1
3
=
3
7
3
= 3.
3
7
=
9
7
Exercise:1.5
1.
푎+1
푎3 −
푎+2
푎2 +
푎+3
푎
2.
5푥3
7푦4 .
21푦2
10푥2
3.
9푥5
36푥2
4.
5−푎
푎2−25
5.
10푎2−29푎+10
6푎2−29푎+10
÷
10푎2−19푎+6
12푎2−28푎+15
6.
1
푥+ℎ
−1
푥
ℎ
7.
푥6−7푥3−8
4푥2−4푥−8 ÷ (2푥2 + 4푥 + 8)
8.
푎
−푏
푏
푎
푎
+푏
푏
푎
9.
푡2−2푡−15
푡2−9
.
푡2−6푡+9
12−4푡
10.
푎−1+푏−1
푎−2−푏−2
14. 1.6 Rational Exponents
We defined 푎푛 if n is any integer (positive, negative or zero). To define a power of a
where the exponent is any rational number, not specifically an integer. That is, we wish to
1
푚
attach a meaning to 푎
⁄푛 푎푛푑 푎
⁄푛, where the exponents are fractions. Before discussing
fractional exponents, we give the following definition.
Definition
The 푛푡ℎ root of a real number
If n is a positive integer greater than
1 푎푛푑 푎 푎푛푑 푏 are real number such that
푏푛 = 푎, then b is an 푛푡ℎ root of a.
Examples 1:
1) 2 is a square root of 4 because 22 = 4
2) 3 is a fourth root of 81 because 34 = 81
3) 4 is a cube root of 64 because 43 = 64
Definition
.
The principal 푛푡ℎ root of a real number. If n is a
positive integer greater than 1, a is a real number,
and √푎 푛 denotes the princial 푛푡ℎ root of a, then
If a>0, √푎 푛 is the positive 푛푡ℎ root of a.
If a<0, and n is odd, √푎 푛 is the negative 푛푡ℎ
The symbol √ is called a radical sign. The entire expression √푎 푛 is called a radical,
where the number a is the radicand and the number n is the index that indicates the
order of the radical.
Examples 2:
1) √4 = 2
2) √81 4 = 3
root of a.
√0 푛 = 0
15. 3) √64 3 = 4
Definition
If n is a positive integer greater than 1, and a is
a real number, then if √푎 푛 is a real number
⁄
Examples 3:
1
⁄2 = √25 = 5
1) 25
1
⁄3= √−8 3 = −2
2) −8
1
81
3) (
1
81
4 =
)1/4=√
1
3
Definition
Examples 4:
3
⁄2=(√9)3=33=27
1) 9
2
⁄3 = (√8 3 )2=22 = 4
2) 8
4
⁄3 = (√−27 3 )4=(-3)4=81
3) −27
It can be shown that the commutative law holds for rational exponents, and therefore
(푎푚)1/n=(푎
1
⁄푛)m
From which it follows that 푛√푎푚 = ( √푎 푛 )m
The next theorem follows from this equality and the definition of 푎
푚
⁄푛
Theorem 1
푎
1
⁄푛 = √푎 푛
If m and n are positive integers that are
relatively prime, and a is a real number,
then if √푎 푛 is a real number
푎
푚
⁄푛 = (√푎 푛 )m ⇔ 푎
푚
⁄푛 = (푎
1
⁄푛)m
If m and n are positive integrers that are
relatively prime, and a is a real number,
then if √푎 푛 is a real number
푎
푚
⁄푛 = 푛√푎푚 ⇔ 푎
푚
⁄푛 = (푎푚)1/n
16. Examples 5:
Theorem 1 is applied in the following:
3
⁄2=√93=729 =27
1) 9
2
⁄3 = √8 3 2=√64 3 = 4
2) 8
4
⁄3 = (√−27 3 )4=√531441 3 =81
3) −27
Observe that 푎
푚
⁄푛 can be evaluated by finding either (√푎 푛 )m or 푛√푎푚. Compare example 4 and 5
and you will see the computation of ( √푎 푛 )m in example 4 is simpler than that for 푛√푎푚 in example 5.
The laws of positive-integer exponents are satisfied by positive-rational exponents with one
exception: For certain values of p and q, (ap)q≠apq for a<0. This situation arises in the following example.
Examples 6:
1) [(-9)2]1/2=811/2=9 and (-9)2(1/2)=(-9)1=-9
Therefore [(-9)2]1/2≠(-9)2(1/2).
2) [(-9)2]1/4=811/4=3 and (-9)2(1/4)=(-9)1/2 (not a real number)
Therefore [(-9)2]1/4≠(-9)2(1/4).
The problems that arise in example 6 are avoided by adopting the following rule: If m and n are
positive even integers and a is a real number, then (푎푚)1/n=│a│m/n
A particular case of this equality occurs when m=n. We then have (푎푛)1/n=│a│ (if n is a positive
even integer) or, equivalently, 푛√푎푛 = │a│ (if n is even)
If n is 2, we have √푎2 = │a│
Examples 7:
1) [(-9)2]1/2=│-9│=9
2) [(-9)2]1/4=│-9│2/4=91/2=3
Definition
If m and n positiv e integer that are
relatively prime and a is a real number and
a≠0, then if √푎 푛 is a real number.
푎
−푚
⁄푛 =
1
푎
푚
⁄푛
19. 2.2 Appplication of Linear Equations
In many applications of algebra, the problems are stated in words. They
are called word problems, and they give relatiomships between known numbers
and unknown numbers to be determined. In this section we solve word
problems by using linear equations. There is no specific method to use.
However, here are some steps that give a possible procedurefor you to follow.
As you read through the examples, refer to these steps to see how they are
applied.
1. Read the problem carefully so that you understand it. To gain
understanding, it is often helpful to make a specific axample that
involves a similar situation in which all the quatities are known.
2. Determine the quantities that are known and those that are
unknown. Use a variable to represent one of the unknown
quantities inthe equation you will obtain. When employing only one
equation, as we are in this section, any other unknown quantities
should be expressed in terms of this one variable. Because the
variable is a number, its definition should indicate this fact. For
instance, if the unknown quantity is a length and lengths are
mesured in feet, then if x is a variable, x should be defined as the
number of feet in the length or, equivalently, x feet is the length. If
the unknown quuantity is time, and time is measured in seconds,
then if t is the variable, t should be defined as the number of
seconds in the time or, equivalently, t seconds is the time.
3. Write down any numerical facts known about the variable.
4. From the information in step 3, determined two algebraic
expressions for the same number and form an equation from them.
The use of a table as suggested in step 3 will help you to discover
equal algebraic expressions.
5. Solve the equation you obtained in step 4. From the solution set,
write a conclusion that answers the questions of the problem.
6. It is important to keep in mind that the variable represents a
number and the equation involves numbers. The units of
measurement do not appear in the equation or its solution set.
7. Check your results by determining whether the condition of the
word problem are satisfied. This check is to verify the accuracy of
the equation obtained in step 4 as well as the accuracy of its
solution set.
![If a=b then a+c=b+c
5. Multiplication Law of Equality
If a=b then a.c=b.c
Axioms for Addition and Multiplication
Let a, b, c, d, ∈ R
1) A. Closure property for addition
a+b ∈ R
Examples:
1) 3+3=6
2) 7+(-4)=3
3) -8+4=-4
B. Closure property for multiplication
a.b ∈ R
Examples:
1) 3(7)=21
2) -8(3)=-24
3) 0.11=0
2) A. Commutative prroperty for addition
a+b=b+a
Examples:
1)
1
2
+ 7 = 7 +
1
2
2) 0.3 + (−
5
6
) = −
5
6
+ 0.3
3)
1
3
+ 21 = 21 +
1
3
B. Commutative prroperty for multiplication
a.b=b.a
Examples:
1)
4
5
4
5
(22) = 22 (
)
2) 6.3=3.6
3)
10
9
10
9
(−25) = −25 (
)
3) A. Associative property for addition
(a+b)+c=a+(b+c)
Examples:
1) (3+7)+0.4=3+(7+0.4)
2) (0.36+89)+
1
2
= 0.36 + (89 +
1
2
)
3
5
3) (
+ 0.8) +
3
8
=
3
5
+ (0.8 +
3
8
)
B. Associative property for multiplication
(a.b).c=a.(b.c)
Examples:
1) (3.x).y=3.(x.y)
2) [5(7)] 1
4
1
4
= 5 [7 (
)]
3) [3푥(6푥)]]5 = 3푥[6푥(5)]
4) Identity property for multiplication](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)