Section 1.4

Objectives: Use the properties of exponents

This section is prepared by M. Zalzali

Positive Integer Exponents

Repeated Form and Exponent form

Exponents are used as a short-hand form for repeated multiplication.

Instead of writing

aaaaa We write

5aRepeated Form

Exponent Form

Note “a” has been repeated 5 times, that’s why we wrote

5a a5 is read “ a to the fifth power’

a = Base, 5 = Power,

Example 1Fill out the table with the correct answer

(3x-1)(3x-1)

(2x+9)(2x+9)(2x+9)

(2x)(2x)(2x)

b6

Exponent FormRepeated Form

7777 47

32x

bbbbbb

213 x 392 x

Rules and Properties ofPositive Integer Exponents

For any real number a and natural numbers m and n.

nmnm aaa

Example 1

First property

xyxyivii

bbiiii

)33)

66)1010)155

5225 710

203

76b

2xy

Simplify each expression. ( Write in exponent form )

…Simplifying Expressions

Note: When we multiply two expressions, we have to take into consideration the associative and commutative properties of multiplication when needed.

Example 2 Simplify the product

25 62 xx 2562 xx 2512 x712x

Class Exercise Simplify the product

33232

24

122)54)

32)35)

cddciiiabbaii

xxivxxi615x

4320 ba

26x6324 dc

…

For any real number a 0, a ,and natural numbers m and n.

m > n

nmn

m

aa

a Second Property

Example 3 Simplify the expressions

nm

nmd

ba

bab

y

yc

x

xa

2

53

43

68

5

14

2

5

5

25

7

35

3x

255 ba

9y

45mn

…

For any real number a and natural numbers m and n.

Third Property mnnm aa

Example 4Simplify the expressions

32

82

53

10c

ab

xa 53x 15x16a

610

…

For any real number a and b ,

Fourth Property mmm baab

Example 5 Simplify the expressions

(a)

(b)

(c)

332

4

5

2

10

qp

a

xy 55 yx

4410 a

333232 qp968 qp

and natural number m

…

For any real number a and b, 0b and natural number m

Fifth Property m

mm

b

a

b

a

Example 6 Simplify the expressions

(a)

(b)

4

3

32

2

b

a

y

x

3

32

y

x3

6

y

x

4

442

b

a4

416

b

a

…

Class Exercise Simplify the expressions

2

5

43

3

2

2

5

c

bab

y

xa

10

86

6

3

4

125

:

c

bab

y

xa

Answers

…For any real number a 0, a

Zero Exponent 10 a

Example 7 Simplify the expressions

010a

06xb 616

032bac

1

1

Home Work

Do the selected exercises from the syllabus

Section 1.4

The end