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