Chapter
p
10: Exponential
p
functions
10.1 INTEGER EXPONENTS
10 2 FRACTIONAL EXPONENTS
10.2
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Integer
ƒ “The numbers”
ƒ Number that has no fractions.
fractions Can be
written without decimal or rational
component.
t
ƒ …— 3, — 2, — 1, 0, 1, 2, 3…
ƒ “…” means “continuing on in the same
pattern without end”
end
Exponent
ƒ The number of times the base is taken as
ƒ
ƒ
ƒ
ƒ
a factor
Don’t apply exponent to values not part of
base
Base includes numbers or variables
attached to the exponent directly
There may be factors of the exponential
that are not part of the base
W t h for
Watch
f ( ) or nott
Negative exponent
ƒ May or may not be a negative number
ƒ Solve by using quotient rule of exponents
3
b
3− 5
−2
=b =b
5
b
Negative exponent
ƒ Solve by using quotient rule of exponents
3
b
3− 5
−2
=b =b
5
b
ƒ Implies moving base across fraction bar
because the two are equivalent
3
b
bbb
1
1
=
=
= 2
5
b
bbbbb bb b
Can only combine LIKE bases
−4
b
1
− 4 −3
−7
=
b
=
b
=
3
7
b
b
−4
b
1
=
3
4 3
c
bc
Simplify Expressions with Exponents
ƒ No parentheses
ƒ Any values are combined into single value
without exponents
ƒ Any variable appears only a single time
ƒ Every exponent is positive
No parentheses
ƒ Rule:
à Multiply the exponent inside by the one outside
(b )
−3 4
=b
−12
1
= 12
b
ƒ Outside exponent means the base, inside,
is the factor to be taken that many times
(b ) = (b )(b )(b )(b ) = b
−3 4
−3
−3
−3
−3
−12
1
= 12
b
No parentheses
ƒ Deal with the outside negative first
(2b )
−5 −3
=
1
(2b )
−5 3
No parentheses
ƒ Multiply the exponents inside by the one
outside
1
(2b )
−5 3
1
= 3 ( −5)3
2b
Values reported without
exponents
1
1
= −15
3 −15
2b
8b
15
1
b
=
−15
8b
8
ƒ And no negative exponents
Can be very tedious when
complicated
ƒ Steps are not hard
ƒ Keep track of where you are
⎛ 18b c
⎜⎜ −3 2
⎝ 6b c
−4 7
⎞
⎟⎟
⎠
−4
ƒ Reduce inside first
(
= 3b
)
− 4 − ( −3) 7 − 2 −4
c
(
)
−1 5 −4
= 3b c
Now deal with ( )
ƒ Multiply each inside exponent by the
outside exponent
(3b c )
−1 5 −4
−4
=3 b
( −1)( − 4 )
c
5(− 4 )
Then deal with negative
exponents
ƒ Any base with a negative exponent needs
to be moved across the fraction bar
−4
4 − 20
3 bc
4
4
b
b
= 4 20 =
20
3 c
81c
ƒ And the numerical exponent needs to
calculated
Scientific notation
ƒ Move decimal point to have a single digit
to left of it
ƒ Multiply by power of 10 to make it
equivalent
i l t expression
i
ƒ Common error is to raise number to
power instead of 10 to power
7 x 103
7 x 10-3
ƒ 7 x 1000 = 7,000
ƒ Move decimal point three places to right
ƒ 7 x 1/1000 = 0.007
ƒ Move decimal point three places to left
ƒ Think about a number line!!
845 000 000
845,000,000
0 0000382
0.0000382
ƒ 8.45 x 108
ƒ 8 from how many decimal places
ƒ 108 is a very large number…
ƒ 3.82 x 10-5
ƒ 5 from how many decimal places
ƒ 10-5 is a very small number…
778 000 000
778,000,000
0 000012
0.000012
ƒ 7.78 x 108
ƒ 8 from how many decimal places
ƒ Did you apply the correct sign of exponent
ƒ 108 is a very large number…LOOK!!
ƒ 1.2
1 2 x 10-5
ƒ 5 from how many decimal places
ƒ 10-5 is a very small number…LOOK!!
Rational exponents:
fractions in exponent!!
ƒ Follow all the same rules for exponents
ƒ Do not assume fractional exponent is a
fractional number!!
ƒ Denominator of exponent means a root
Rules for
exponents:
page 594
ƒ The last rule:
doesn t matter if
doesn’t
n gets switched
with m
ƒ Important note
for the rational
exponents
b b =b
m
n
m+n
m
b
m−n
=b
n
b
n
n n
(bc ) = b c
n
b
⎛b⎞
⎜ ⎟ = n
c
⎝c⎠
(b )
m n
n
=b
m ⋅n
/ )
Rational exponent 9((1/2)
ƒ Note you can write 9 as 32
ƒ Apply rule for nested exponents
9
( 12 )
( 12 )
( )
= 3
2
=3
( 2)
2 1
=3 =3
1
Watch out for negative base
ƒ If root is odd, retain the negative
(− 64)
1
3
[
= (− 4 )
3
]
1
3
= −4
ƒ If root is even, it is not a real number
(− 16)
ƒ But
1
4
[
= (− 2 )
2 = 16
4
4
]
1
4
=2
so above is not true