EXPONENTS




Basic Exponent Laws
Negative Exponents
Exponential Expressions
Exponential Equations
1
BASIC EXPONENT LAWS
Recap of the meaning of exponents
E.g.
 a)  2 3
vs
 24
  2 . 2 .2
  2 .2 .2 .2
 8
 16
 b)  (2) 2
 ( 2)( 2)
 4
( )even power
= positive answer
Powers of -1
vs
 (2) 5
 (2)( 2)( 2)( 2)( 2)
 32
( )odd power
= negative answer
2
Reminder!
• 2a = a + a = 2 x a
• a2 = a . a
• (a +
2
b)
2
=a
+ 2ab +
≠ (a2 +b2)
2
b
3
Laws
an – n
E.g. a)
= a0 = 1
24
24
= 24-4
= 20
=1
b)
c)
d)
60 = 1
x³y0 = x³.1 = x³
4x0 + (4x)0 = 4.1 + 1 = 5
4
an = a . a . a . a ... to n factors
(a > 0
and a ≠ 1, n ε N)
E.g. a) 25 = 2.2.2.2.2
b) x3 = x.x.x (not x+x+x = 3x)
c) 4x²y³ = 4.x.x.y.y.y
Basic Laws of Exponents
5
a n . a m = a m+n
Eg: a) 23 . 22
= 23 + 2
= 25
= 32
b) x4y6.3x5y4
= 3x9y10
Multiplying Exponents with the Same Base
6
(ambn)p
= am.p bn.p
E.g. a) (23)4
= 23x4
= 212
b) (2x6y5)4
= 24x6x4 y5x4
= 16x24y20
c)
8
 
 3
2
 82 
  2 
3 
64

9
7
a n ÷ a m = a n–m
E.g. a) 35 ÷ 33
= 35 – 3
= 32
= 9
b)
34x7 y3 ÷ 8x5 y8
= 4 x7–5 y3–8
= 4x2y-5
Dividing Exponents with
the Same Base
8
Exercise
9
NEGATIVE EXPONENTS
So, if the exponent is NEGATIVE in
the NUMERATOR => move the
factor to the denominator and make
the exponent POSITIVE
So, if the exponent is NEGATIVE in the
DENOMINATOR => move the factor to
the numberator and make the exponent
POSITIVE
10
Exercise
Quiz: Exponents
11
EXPONENTIAL EXPRESSIONS
E.g. a)
x
x 3
1 x
5 .5 .5
0 x
5 .5
Same base –
apply exponent
laws!
5
x  ( x 3)  (1 x )  0  (  x )
5
x  x 31 x  0  x
5
2 x2
12
E.g.
b) Simplify
x
x 3
8 .9
x 1  x
6 .3
Can’t use
Exponent Laws
…. why? ….
Different Bases! So, we make the bases the
same by prime factorizing ….
Prime: 2; 3; 5; 7; 11; 13; 17; 19; 23 …
13
x
x 3
8 .9
x  x 1
6 .3
2 x 3
3 x
(2 ) .(3 )

x  x 1
(2.3) .3
2 x 6
3x
2 .3
 x x  x 1
2 3 .3
2
3x x
.3
 2 .3
2x
2 x  6  x  (  x 1)
2 x 5
14
Challenge!
x
x 3
1 x
8 .6 .9
x 1  x
16 .3
3 x
x 3
2 1 x
(2 ) .(2.3) .(3 )

4 x 1  x
(2 ) .3
x 3
3x
x 3
2 .2 .3 .3

4 x4  x
2 .3
2
3 x  x  3 4 x  4
 2 .3
.3
1
2

3
22 x
x  3 2  2 x  x
15
Challenge!
9 1
x
3 1
x
x
(3  1)(3  1)

x
(3  1)
x
 (3  1)
x
16
Exercise
Simplify:
1.
9x
27 x 1
2.
3 2  7 x  2  32 x
7 5  34 x
n 1
n 1
2
.
3

4
.
3
3.
4.
5n  5n  2
5n 5
17
EXPONENTIAL EQUATIONS
Solve for x:
1.
2 2 x  16
22 x  24
2x  4
x2
2.
2.32 x 1  18
32 x 1  9
32 x 1  32
2 x 1  2
3
x
2
18
Solve for x: 3.
82 x
1
2 x 1
16
( 23 ) 2 x
1
4 2 x 1
(2 )
26 x
1
8 x4
2
19
Solve for x: 4.
3 x  3 x 1  12
3 x  3 x.31  12
1
3 (1  3 )  12
x
1
3  12  (1  3 )
1
x
3  12  (1  )
3
3
x
3  12 
4
3x  9
x
3 x  32
x2
20
Solve for x: 5.
1
2
1
4
x  7 x  18  0
1
4
1
4
( x  9)( x  2)  0
1
4
1
4
x  9  0 _ or _ x  2  0
1
4
1
4
x  9 _ or _ x  2
1
4 4
No _ Solution _ or _( x )  24
x  16
21
Exercise
Solve for x:
1.
7 x 1  49
2.
4.3 x  108
3.
2.5  x 1  250  0
4.
5
2 x 1
1

25
22