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 𝑅. 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

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(22) = 22 (

4

5)

2) 6.3=3.6

3) 10

9(−25) = −25 (

10

9)

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) (3

5+ 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= 5 [7 (

1

4)]

3) [3𝑥(6𝑥)]]5 = 3𝑥[6𝑥(5)] 4) Identity property for multiplication

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

𝑎.1

𝑎= 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

1.2 Exponents and Radicals In the expression 𝛼𝑛, α is the base and 𝘯 is the exponent. The expression 𝛼𝑛 means that the value α is multiplied 𝘯 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)

𝑎0 = 1 Examples:

1) (6𝑎0 + 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. α𝘯= αm+n where m and n are integers. Examples:

1) α5.α4=𝑎5+4 = 𝑎9 2) 4𝑥𝑦2(2𝑥2𝑦2) = 8𝑥1+2𝑦2+2 = 8𝑥3𝑦4 3) 𝑥𝑎+3. 𝑥𝑎+4 = 𝑥2𝑎+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:

For any real numbers a and b (b≠0),

(𝑎

𝑏)𝑛 =

𝑎𝑛

𝑏𝑛

where n is any integer. Examples:

1) (𝑎2

𝑏3)2 =

𝑎4

𝑏6

2) (3

4)3 =

33

43=

27

64

3) (𝑥

𝑦+2)2=

𝑥2

(𝑦+2)2=

𝑥2

𝑦2+4𝑦+4

Theorem 6: For any real numbers a(a≠0),

𝑎𝑚

𝑎𝑛 = 𝑎𝑚−𝑛

where m and n are integers. Examples:

1) 𝑎7

𝑎5= 𝑎7−5=𝑎2

2) 𝑥3𝑦4𝑧5

𝑥𝑦𝑧= 𝑥3−1𝑦4−1𝑧5−1 = 𝑥2𝑦3𝑧4

3) 𝑥4𝑦4

𝑥4𝑦4= 𝑥4−4𝑦4−4 = 𝑥0𝑦0 = 1(1) = 1

Theorem 7:

For any real numbers a(a≠0),

𝑎−𝑛 =1

𝑎𝑛

Where n is any positive integer. Examples:

1) 3𝑥3𝑦−2=3𝑥3

𝑦2

2) (4𝑥2𝑦)−2 =1

(4𝑥2𝑦)2=

1

8𝑥4𝑦2

3) (𝑥2 + 𝑦)−2 =1

(𝑥2+𝑦)2=

1

𝑥4+𝑦2

Exercises 1.2.1 Simplify and express the following expressions with positive and negative integrals only.

1. 50

2. 10𝑚4

30m

3. 16𝑏4𝑐

−4𝑏𝑐3

4. 𝑦3. 𝑦4 5. (5𝑥𝑦) 6 6. (𝑎𝑏) 3 7. (𝑥3𝑦2)3

8. [(−5)2]2

9. 𝑥5𝑦6

𝑥𝑦=

10. 𝑎7

𝑎3

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 𝑥1

2 is the same as √2 (read as square root of 2), and 𝑥2

3 is the same as

√𝑥23 (read as cube root of x squared). The expression √𝑎𝑚𝑛 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.

𝑎𝑚𝑛

= √𝑎𝑚𝑛

Laws of Radicals Theorem 1: For any real numbers a,

√𝑎𝑛 = 𝑎𝑛

Examples:

1) √42 = 4

2) √(𝑥2𝑦)33 =𝑥2𝑦

3) √333=3

Theorem 2: For any real numbers a,and b.

√𝑎𝑛 . √𝑏𝑛 =√𝑎𝑏𝑛 Examples:

1) √3. √3 = √3.3 = √9=3

2) √4. √3 = √4.3 = √12

3) √𝑎. √𝑏 = √𝑎. 𝑏 Theorem 3: For any real numbers a,and b, (b≠0)

√𝑎𝑛

√𝑏𝑛= √

𝑎

𝑏

𝑛

Examples:

1) √𝑎3

√𝑏3 = √𝑎

𝑏

3

2) √4

√5= √

4

5

3) √𝑥4

√𝑦4 = √𝑥

𝑦

4

Theorem 4: For any real numbers a ,

√𝑎𝑚𝑛 = √ √𝑎𝑛𝑚= √ √𝑎𝑚𝑛

Examples:

1) √646 = √√643

= √83 = 2

2) √164 = √√162

= √42 =2

3) √1003 = √1003 =√100 = 10 Theorem 5:

For any real numbers a

k √𝑎𝑘𝑚𝑛= √𝑎𝑚𝑛

Examples:

1) √246= √22.22.3

= √223= √43

2) √936= √93.13.2

= √92 =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) 𝑦√𝑥3𝑦 − √𝑥3𝑦3 + 𝑥√𝑥𝑦3 = 𝑦√𝑥2. 𝑥𝑦 − √𝑥2. 𝑥. 𝑦2. 𝑦 + 𝑥√𝑥. 𝑦2. 𝑦 = 𝑥𝑦√𝑥𝑦 −

𝑥𝑦√𝑥𝑦 + 𝑥𝑦√𝑥𝑦 = 𝑥𝑦√𝑥𝑦

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) √𝑥𝑦3 .√𝑥2𝑦3 . √𝑥𝑧3 =√𝑥𝑦. 𝑥2𝑦. 𝑥𝑧3 = √𝑥4𝑦2𝑧3 = 𝑥 √𝑥𝑦2𝑧3

3) √163 ÷√−23 =√16 ÷ (−23 )=√−83 = −2

Exercise: 1.2.2

Simplify and solve.

1. (5√2)(3√6)

2. (3𝑎√4𝑥23)(4√3𝑥𝑦3 )

3. 4√9

16

4. √2(3+√3)

5. 5√2+3√2

6. √18 − 2√27 + 3√3 − 6√8

7. √16𝑏 + √4𝑏

8. −12√24

3√2

9. √8 + √50

10. 4√𝑥7𝑦10

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 x 2+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 coeffici ents 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) 𝑥𝑚.𝑥𝑛 = 𝑥𝑚+𝑛

2) 𝑥−2.𝑥2=𝑥0 = 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.

𝑥𝑚

𝑥𝑛 = 𝑥𝑚−𝑛 𝑎𝑛𝑑 𝑥−𝑛 =1

𝑥𝑛

Examples:

1) 𝑥5

𝑥2= 𝑥3

2) 𝑥−5=1

𝑥5

3) Divide 7𝑥2 − 5𝑥 𝑏𝑦 𝑥 𝑥 is the divisor and 7𝑥2 − 5𝑥 as the dividend, we have

7𝑥2−5𝑥

𝑥=

7𝑥2

𝑥-

5𝑥

𝑥=7𝑥 − 5

Exercise: 1.3

1. (5𝑥 − 1) + (10𝑥2 + 7𝑥) 2. (20𝑥2 + 2) + (15𝑥2 − 8) + (3𝑥2 − 4) 3. (𝑥2 + 𝑦2 + 8) + (4𝑥2 − 2𝑦2 − 9) 4. (−3𝑥2 + 5𝑦 − 4𝑥𝑦 + 𝑦2) 𝑓𝑟𝑜𝑚(2𝑥2 − 4𝑦 + 7𝑥𝑦 − 6𝑦2)

5. 2𝑥2 + 6𝑥 + 5 𝑎𝑛𝑑 3𝑥2 − 2𝑥 − 1 6. (𝑥 + 2)(𝑥2 − 2𝑥 + 3) 7. 𝑎𝑏(2𝑎 + 1)

8. 𝑥2−3𝑥−10

𝑥+2

9. 𝑥6+2𝑥4+6𝑥−9

𝑥3+3

10.

1.4 Factoring Polynomials

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) 4𝑎2𝑏𝑐3

6𝑎𝑏3𝑐4=

2.2.𝑎.𝑎.𝑏.𝑐.𝑐.𝑐

2.3.𝑎.𝑏.𝑏.𝑏.𝑐.𝑐.𝑐.𝑐=

2𝑎

3𝑏2𝑐

2) 𝑥2+2𝑥𝑦+𝑦2

𝑥2−𝑦2=

(𝑥+𝑦)(𝑥+𝑦)

(𝑥+𝑦)−(𝑥−𝑦)=

𝑥+𝑦

𝑥−𝑦

3) 𝑥3+8𝑦3

4𝑥+8𝑦=

𝑥+2𝑦(𝑥2−2𝑥𝑦+4𝑦2

4(𝑥+2𝑦)=

𝑥2−2𝑥𝑦+4𝑦2

4

1.5.2 Multiplying and Dividing Rational Expressions

In multiplication if 𝑝

𝑞𝑎𝑛𝑑

𝑟

𝑠 are rational expressions and q and s are real numbers not equal to 0,

then 𝑝

𝑞.

𝑟

𝑠=

𝑝𝑟

𝑞𝑠.

Examples:

1) 4

3.

1

5=

4

15

2) 𝑐

𝑎2−𝑏2. (𝑎 + 2𝑏)(𝑎 − 𝑏)

=𝑐

(𝑎 + 𝑏)(𝑎 − 𝑏). (𝑎 + 2𝑏)(𝑎 − 𝑏)

=𝑐(𝑎+2𝑏)

𝑎+𝑏

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

18𝑜𝑟

10

9

2) 8

7÷ 3 =

8

7.

1

2=

8

14𝑜𝑟

4

7

3) 𝑦2−16

𝑦−5÷

2𝑦−8

𝑥𝑦−5𝑥=

(𝑦−4)(𝑦+4)

𝑦−5.

𝑥(𝑦−5)

2(𝑦−4)=

𝑥𝑦+4𝑥

2

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) 3𝑥 − 2𝑦 +2𝑥2−𝑦2

𝑥+𝑦=

3𝑥(𝑥+𝑦)−2𝑦(𝑥+𝑦)+2𝑥2−𝑦2

𝑥+𝑦=

3𝑥2+3𝑥𝑦−2𝑥𝑦+2𝑦2+2𝑥2−𝑦2

𝑥+𝑦=

5𝑥2+𝑥𝑦−3𝑦2

𝑥+𝑦

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 𝑜𝑟 4

2) 3

2+1

3

=3

6+1

3

=37

3

= 3.3

7=

9

7

Exercise:1.5

1. 𝑎+1

𝑎3−

𝑎+2

𝑎2+

𝑎+3

𝑎

2. 5𝑥3

7𝑦4.

21𝑦2

10𝑥2

3. 9𝑥5

36𝑥2

4. 5−𝑎

𝑎2−25

5. 10𝑎2−29𝑎+10

6𝑎2−29𝑎+10÷

10𝑎2−19𝑎+6

12𝑎2−28𝑎+15

6. 1

𝑥+ℎ−

1

𝑥

ℎ

7. 𝑥6−7𝑥3−8

4𝑥2−4𝑥−8÷ (2𝑥2 + 4𝑥 + 8)

8. 𝑎

𝑏−

𝑏

𝑎𝑎

𝑏+

𝑏

𝑎

9. 𝑡2−2𝑡−15

𝑡2−9.

𝑡2−6𝑡+9

12−4𝑡

10. 𝑎−1+𝑏−1

𝑎−2−𝑏−2

1.6 Rational Exponents We defined 𝑎𝑛 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

attach a meaning to 𝑎1

𝑛⁄ 𝑎𝑛𝑑 𝑎𝑚

𝑛⁄ , where the exponents are fractions. Before discussing fractional exponents, we give the following definition. Definition

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 symbol √ is called a radical sign. The entire expression √𝑎𝑛 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) √814 = 3

The 𝑛𝑡ℎ root of a real number

If n is a positive integer greater than

1 𝑎𝑛𝑑 𝑎 𝑎𝑛𝑑 𝑏 are real number such that

𝑏𝑛 = 𝑎, then b is an 𝑛𝑡ℎ root of a.

The principal 𝑛𝑡ℎ root of a real number. If n is a

positive integer greater than 1, a is a real number,

and √𝑎𝑛 denotes the princial 𝑛𝑡ℎ root of a, then

If a>0, √𝑎𝑛 is the positive 𝑛𝑡ℎ root of a.

If a<0, and n is odd, √𝑎𝑛 is the negative 𝑛𝑡ℎ

root of a.

√0𝑛

= 0

3) √643 = 4 Definition

⁄

Examples 3:

1) 251

2⁄ = √25 = 5

2) −81

3=⁄ √−83 = −2

3) (1

81)1/4=√

1

81

4=

1

3

Definition Examples 4:

1) 93

2⁄ =(√9)3=33=27

2) 82

3⁄ = (√83 )2=22 = 4

3) −274

3⁄ = ( √−273 )4=(-3)4=81

It can be shown that the commutative law holds for rational exponents, and therefore

(𝑎𝑚)1/n=(𝑎1

𝑛⁄ )m

From which it follows that √𝑎𝑚𝑛 = ( √𝑎𝑛 )m

The next theorem follows from this equality and the definition of 𝑎𝑚

𝑛⁄

Theorem 1

If n is a positive integer greater than 1, and a is

a real number, then if √𝑎𝑛 is a real number

𝑎1

𝑛⁄ = √𝑎𝑛

If m and n are positive integers that are

relatively prime, and a is a real number,

then if √𝑎𝑛 is a real number

𝑎𝑚

𝑛⁄ = ( √𝑎𝑛 )m ⇔ 𝑎𝑚

𝑛⁄ = (𝑎1

𝑛⁄ )m

If m and n are positive integrers that are

relatively prime, and a is a real number,

then if √𝑎𝑛 is a real number

𝑎𝑚

𝑛⁄ = √𝑎𝑚𝑛 ⇔ 𝑎𝑚

𝑛⁄ = (𝑎𝑚)1/n

Examples 5:

Theorem 1 is applied in the following:

1) 93

2⁄ =√93=729 =27

2) 82

3⁄ = √83 2=√643 = 4

3) −274

3⁄ = ( √−273 )4=√5314413 =81

Observe that 𝑎𝑚

𝑛⁄ can be evaluated by finding either ( √𝑎𝑛 )m or √𝑎𝑚𝑛 . Compare example 4 and 5

and you will see the computation of ( √𝑎𝑛 )m in example 4 is simpler than that for √𝑎𝑚𝑛 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 (𝑎𝑚)1/n=│a│m/n

A particular case of this equality occurs when m=n. We then have (𝑎𝑛)1/n=│a│ (if n is a positive

even integer) or, equivalently, √𝑎𝑛𝑛 = │a│ (if n is even)

If n is 2, we have √𝑎2 = │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 √𝑎𝑛 is a real number.

𝑎−𝑚

𝑛⁄ =1

𝑎𝑚

𝑛⁄

Example:8

1) 8−2

3⁄ =1

82

3⁄=

1

( √8)23 =1

22=

1

4

2) 8−2

3⁄ = (8−1

3⁄ )2=(1

81

3⁄)2=(

1

2)2=

1

4

3) 𝑥

13⁄

𝑥1

4⁄ =𝑥

13⁄ .

1

𝑥1

4⁄=𝑥

13⁄ . 𝑥

−14⁄ = 𝑥(1

3⁄ )−14⁄ = 𝑥

112⁄

Exercise 1.6

1. 𝑎)811

2⁄ ; 𝑏)271

3⁄ ; 𝑐)6251

4⁄ ; 𝑑)321

5⁄

2. 𝑎)161

2⁄ ; 𝑏)1251

3⁄ ; 𝑐)161

4⁄ ; 𝑑)1000001

5⁄

3. 𝑎)𝑥−3

4⁄ . 𝑥5

6⁄ . 𝑥−1

3⁄ ; 𝑏)𝑦

−34⁄

𝑦3

2⁄)-1/9

4. 𝑎)𝑦1

4⁄ . 𝑦−3

2⁄ . 𝑦−5

8⁄ ;𝑏)𝑥

−35⁄

𝑥−7

10⁄)-1/4

5. (𝑥1

3⁄ − 𝑥−2

3⁄ )(𝑥2

3⁄ − 𝑥−1

3⁄ )

6. (𝑎1

4⁄ − 𝑎1

2⁄ )(𝑎−1

4⁄ + 𝑎−1

2⁄ )

7. 𝑎)2𝑦3

2⁄ − 3𝑦5

2⁄ ; 𝑏)5𝑥−4

3⁄ + 4𝑥5

3⁄

8. 𝑎)6𝑡3

4⁄ + 𝑡7

4⁄ ; 𝑏)4𝑤4

5⁄ − 3𝑤−6

5⁄

9. a)(𝑎3)n/3(𝑎3𝑛)3/n; b)( 𝑥𝑛

2⁄ )-1/2(𝑥−1

2⁄ )-n

10. a)(𝑦4)n/4(𝑦2𝑛)2/n; b)( 𝑡𝑛

3⁄ )-2/3(𝑡−1

3⁄ )-n

Chapter II Equations and Inequalities

2.1 Equations

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.