Transcript p.2 - abu-saba

P.2
• Properties of exponents
• Simplifying exponential expressions
• Converting from Decimal Notation to Scientific Notation
• Converting from Scientific Notation to Decimal Notation
Pg. 25 #26-86 every other even
Note: For the simplifying exponential expressions section, leave your
answers with all positive exponents in them – no negative exponents!
The Negative Exponent Rule:
Examples:
1
1
5  3
5 125
3
When an exponent is negative,
the entire power (base and exponent)
can be moved to the other side of the
fraction. Once moved, the exponent
is written as a positive.
Simplify the power to finish the problem.
1
2

4
 16
2
4
The Zero Exponent Rule:
Anything raised to the zero power becomes the value 1.
Examples:
(-3)0 = 1
(6x4y-3)0 = 1
Simplifying by Grouping Same Base Powers
45∙42
= 4∙4∙4∙4∙4 times 4∙4
A group of five 4s times a group of two 4s is:
4∙4∙4∙4∙4∙4∙4
= 47
(3x2y4)5
= (3x2y4) (3x2y4) (3x2y4) (3x2y4) (3x2y4)
Five groups of (3x2y4)
= 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ x2 ∙ x2 ∙ x2 ∙ x2 ∙ x2 ∙ y4 ∙ y4∙ y4 ∙ y4∙ y4
= 243 x10y20
M5
M3
= M∙M∙M∙M∙M = M∙M∙M∙M∙M = M∙M = M2
M∙M∙M
M∙M∙M
M =1
M
These M pairs cancel; leaving only two Ms in the numerator.
(2∕7)3 = (two sevenths raised to the third power)
= (2∕7)(2∕7)(2∕7)
2∙2∙2 = 8
7∙7∙7
343
Simplify:
A) -35x2y3 =
5x6y-8
Divide -35 by 5 = -7
= -7x2y3
x6y-8
= x∙x
x∙x∙x∙x∙x∙x
= -7y3
x4y-8
= -7y3y8
x4
= -7y11
x4
Two of the x pairs cancel.
The y-8 must move to the numerator and
the exponent becomes positive 8.
Finally, exponential groups of base y
in the numerator can be combined.
Simplify:
1. (2x3y2)4
2. (-6x2y5)(3xy3)
Your final simplified expression should have no negative exponents.
3.
4.
100 x9 y 2
20 x 6 y 4
 5x 
 4
y 
2
In Decimal Notation (how number values are most commonly written), very large
and very small numbers require a long series of zeros to denote.
For example:
0.00000000237
and
1,367,000,000,000
Scientific Notation was invented so that numbers of very large or very small size
could be written with fewer symbols.
Here are the above numbers in scientific notation:
2.37 x 10-9
and
1.367 x 1012
Notice that the very small number on the left (0.00000000237) is associated with
a negative exponential power of ten in its scientific notation (-9).
Conversely, the very large number on the right (1,367,000,000,000 ) is
associated with a positive exponential power of ten in its scientific notation (12).
This correlation always holds true in these conversions.
To Convert from Decimal (Normal) Notation to Scientific Notation:
A)
0.000000408
Take the decimal and move it to the right of the first non-zero digit you pass.
Like this:
0 0000004.08
You moved it seven places right to obtain the number 4.08
Since the value itself (0.000000408) is actually very small, we use negative
exponent in our final answer:
4.08 x 10 -7
B)
120,500
Take the decimal (which is unseen on the far right) and move it to the left until there
is only one non-zero digit left. Like this
120 500.
becomes
1.20500
You moved it five places to the left to obtain the number 1.205
Since the number itself (120,500) is actually very large, we use a positive exponent
in our final answer:
1.205 x 105
To Convert from Scientific Notation to Decimal (Normal) Notation:
C) 7.035 x 106
This exponent is positive (6), which means that the actual number is very large. So,
we should move the decimal to the RIGHT to make the value larger. The
exponent tells us how many places right we should go: six
Starting with 7.035 and moving right six places we get 7 035_ _ _ .
Moving the decimal in this way leaves us with empty spaces. We fill them in with
zeros to obtain our final answer: 7035000
Or, if you prefer: 7,035,000
D) 8.65 x 10-2
This exponent is negative (-2), which means the actual number is very small.
So, we should move the decimal to the LEFT to make the value smaller.
The exponent tells us how many places left we should go: two
Starting with 8.65 and moving left two places we get . _ 865
Again, moving the decimal in this way leaves us with empty spaces. Again, we fill
them in with zeros to obtain our final answer: .0865
Or, if you prefer: 0.0865
Write each number in decimal notation.
5. -2.6 x 109
6. 3.017 x 10-6
Write each number in scientific notation.
7. 5,210,000,000
8. -0.00000006893