10.1 Scientific Notation
10
1 Scientific Notation
10.2 Rational Exponents
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Scientific notation
Scientific notation
• Move decimal point to have a single digit to left of it
• Multiply by power of 10 to make it equivalent expression
• Common error is to raise number to power instead of 10 to power
7 x 10
7
x 103
7 x 10
7
x 10‐33
• 7 x 1000 = 7,000
• Move decimal point three places to right
Move decimal point three places to right
• 7 x 1/1000 = 0.007
• Move decimal point three places to left
Move decimal point three places to left
• Think about a number line!!
845 000 000
845,000,000
•
•
•
•
•
•
0 0000382
0.0000382
8.45 x 108
8 from how many decimal places
8 from how many decimal places
108 is a very large number…
3.82 x 10‐5
5 from how many decimal places
5 from how many decimal places
10‐5 is a very small number…
778 000 000
778,000,000
•
•
•
•
•
•
•
0 000012
0.000012
7.78 x 108
8 from how many decimal places
8 from how many decimal places
Did you apply the correct sign of exponent
108 is a very large number…LOOK!!
1 2 x 10‐5
1.2 x 10
5 from how many decimal places
10‐5 is a very small number…LOOK!!
10 2 Fractional Exponents
10.2 Fractional Exponents
• Fractional exponents are roots
• Follow same rules as integer exponents
Follow same rules as integer exponents
Rational exponents: fractions in exponent!!
• Follow all the same rules for exponents
• Do not assume fractional exponent is a Do not assume fractional exponent is a
fractional number!!
• Denominator of exponent means a root
i
f
Rules for Rules
for
exponents:
page 594
ƒ The last rule: doesn t matter if
doesn’t
matter if
n gets switched
with m
ƒ Important note for the rational
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
Rational
exponent 9(1/2)
• Note you can write 9 as 32
• Apply rule for nested exponents
Apply rule for nested exponents
9
( 12 )
( 12 )
( )
= 3
2
=3
( 2)
2 1
=3 =3
1
b1/n
• Root of b with index of n
• If the denominator is even, b must be positive If the denominator is even, b must be positive
or the root is not real
• If the denominator is odd, b is real, and the If h d
i
i dd b i
l d h
same sign as b
bm/n
•
•
•
•
/
(bm)1/n
(b1/n)m
Same thing
Whichever is more obvious to you
253/2
•
•
•
•
/ )3
(251/2
[(52)1/2]3
53
125
(4x6)3/2
•
•
•
•
•
•
/ )(x18/2
/ )
(43/2
(43) 1/2(x12)1/2
(41/2)3(x6/2)3
23∙x3∙3
8x9
Yes, you can reduce in the fractional exponent, I recommend it
Nested quotient with rational exponent
⎛ 32 x
⎜⎜ 12
⎝ x
2
⎞
⎟⎟
⎠
2
5
• Yes, you can reduce the base of x before you 2
do any thing else, I recommend it
y
g
,
5
32
⎛
⎞
⎜ 10 ⎟
• Apply exponent to all factors
⎝x ⎠
• Yes, you can reduce the fractional
Y
d
h f
i
l
2
2
exponent, I recommend it
5
2
5
5
2
32
2
• While you are at it, why
20
4
4
5
x
x
x
not write 32 as exponent
not write 32 as exponent
• and use nested rule there too
( )
Rules for Rules
for
exponents:
page 594
ƒ The last rule: doesn’t matter if n gets matter
if 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
Watch out for negative base
Watch out for negative base
• If root is odd, retain the negative
f
dd
h
(− 64)
1
3
[
= (− 4 )
3
]
1
3
= −4
• If root is even, it is not a real number
If root is even it is not a real number
(− 16)
• But
1
4
2 = 16
4
[
= (− 2 )
4
]
1
4
=2
so above is not true
Always change numbers into exponent expressions when possible
•
•
•
•
81 = 34
64 = 2
64 26
625 = 54
216 = 63
Rules for Rules
for
exponents:
page 594
ƒ The last rule: doesn’t matter if n gets matter
if 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