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Scientific Notation
Remember….
Scientific notation is a single digit coefficient times a
power of ten.
Standard  scientific
430, 000  4.3 * 105
0.0000057  5.7 * 10-6
Positive exponent for large numbers
Negative exponent for small numbers
To convert from scientific notation to standard (regular number) look at the power
of ten. The exponent tells you which direction to move the decimal, and how far.
Negative exponent moves the decimal to the left so the number gets smaller
Positive exponent moves the decimal to the right so the number gets larger
You try! Convert each number to the missing form
Standard Form
1,230,000,000
Scientific Notation
7.459 * 10-5
2 * 108
.0000009
8.05 * 107
-0.0034
63,910,000
3.331 * 10-4
-2.081 * 105
.0000147
Rewrite the following numbers in scientific notation
a. 56.8 * 10-5
b. 0.123 * 108
c. 60 * 10-3
d. 802 * 106
Multiplying and Dividing in Scientific Notation
Multiplying:
(1.5 * 105) (8 * 107)
Using the commutative property of multiplication, we can rearrange the factors in
the problem like:
1.5 * 8 * 105 * 107
Then, using the associative property we can the group the numbers and simplify
like:
(1.5 * 8 ) (105 * 107)
= 12 * (10 * 10* 10* 10* 10)( 10* 10* 10* 10* 10* 10* 10)
= 12 * 1012
Since this is not scientific notation we must rewrite it as a single digit times a power
of ten; This is most easily done by changing the number to standard notation and
then back to scientific notation.
12,000,000,000,000  1.2 * 1013
Dividing:
16∗105
4∗ 102
First, we can rewrite the problem as two fractions like:
Then simplify each fraction (division):
4*
Make sure your answer is in scientific notation!
16
4
∗
10∗10∗10∗10∗10
10∗10
4 * 103
You try! Make sure all answers are in scientific notation!
(3.5 * 105) (4 * 103)
(2 * 107) (4 * 104)
30∗1012
48∗1015
3∗ 104
12∗ 1012
105
102
Rational vs. Irrational Numbers
Remember…..
A rational number is any number that can be written as a fraction. This is the
majority of numbers!
An irrational number is a real number that can NOT be written as a fraction. The
most common examples of this are pi (3.14….) and the square root of non perfect
squares like √41.
There are some strange numbers that are not real and we can’t place on a number
line. Examples of this are the square root of a negative number, and dividing by 0.
1) Are all square roots irrational? Provide examples to explain your thinking.
2) Write multiple irrational numbers between 3 and 4.
3) Identify if each number is rational, irrational, or not real.
-4
3
√−8
7
0
78%
0.057
1
83
-3.02
13
7
√150
−√64
√−1
pi
Square Roots
Remember,
When taking the square root of a number, there are TWO roots!
√64 = 8 because 8 * 8 = 64
√64 = −8 because -8 * -8 = 64
When estimation non perfect square roots you can “surround” it with perfect
squares. Then on a number line you can estimate which two integers the square root
is between.
You can tell which half of the number line to estimate on based on which perfect
square it is closer to.
1) List out the perfect SQUARES 1 -20.
2)Can you take the square root of a negative number? Why or why not?
2) What are the two square roots of the following numbers?
25
121
225
3)Estimate the following square roots to the nearest tenth.
√207
√178
√3
√109
√300
Cube Roots
A cube root is the side length of a cube given the volume.
3
√27 = 3 because 3 * 3 * 3 = 27
When estimation non perfect cube roots you can “surround” it with perfect cubes.
Then on a number line you can estimate which two integers the cube root is
between.
You can then tell which half of the number line to estimate on based on which
perfect cube it is closer to.
1) List out the first ten perfect CUBES on your notecard for the quiz.
2) Can you take the cube root of a negative number? Why or why not?
3) Estimate the following cube roots to the nearest tenth.
3
√900
3
√234
3
√20
3
√150
3
√9
3
√612
Order of Operations
Examples: 18 ÷ 32 • √16 + 4 • 52
= 18 ÷ 9 • 4 + 4 • 25 (#3 exponents)
= 2 • 4 + 4 • 25
(#4- division)
= 8 + 4 • 25
(#4-multiplication)
= 8 + 100
(#4-multiplication)
= 108
(#5-addition)
Simplify the expressions using Order of Operations
1) √5 + 11
2)
3√32 + 32
3)
13 – 2 √9
4) √1 + (12 · 4)
5) √(6 − 2) ∗ 52
6) 72 ÷ 9(4) ÷ 5√4
7) (√81 – 8)3 + 3 • 24 + 0 • 52
Graphing Numbers on Number Lines
Number lines are usually marked with rational numbers. On simple number lines, we
1
count by integers. More complex number lines may count by 2’s or tenths. The way the
number line is divided up will depend on the numbers you are plotting on the number
line.
No matter how the number line is set up, you will need the rational approximation of an
irrational number to graph it on the number line.
Change each number to a decimal and approximate its position on the number line.
1) Graph the following numbers on a number line then order from least to greatest.
√9 , 1.5 ,
7
16
3
5
,
, and
√16
16
Order: ____________________________________________________________________
2)
Comparing Numbers
You can compare two numbers written in scientific notation by looking at their
greater power of 10 (exponent). The number with the greater power of 10 will be
the greater number.
If two numbers have the same power of 10, then compare the coefficient to
determine the greater number.
Compare: 8.43×106 and 2.38X108 Because the exponents are different, you know
that the number with the greater exponent is the greater number. 8 is greater than
6. Therefore 8.43×106 < 2.38X108
Compare: 3.2×10-10 and 1.2×10-9 Even though the exponents are negative, because
they are different you still know that the number with the greater exponent is the
greater number. -9 is greater than -10. Therefore 3.2×10-10 < 1.2×10-9
Compare: 5.65×105 and 5.56×105 Because the exponents are the same, you will
have to compare the coefficient to determine the greater number. 5.65 is greater
than 5.56. Therefore 5.65×105 > 5.56×105
As always, a negative number (negative coefficient) will always be smaller than a
positive number. Changing numbers into standard notation is another option.
When in doubt, expand it out!
Compare the following. Write <, >, or = for each blank
3.3×102__________ 3.1×103
5.5×106 _________ 5.51× 106
7.2×10-4__________ 8.9×10-5
8.71×10-3_________4.16×10-3
5.5×106 _________ -5.51× 108
Order from least to greatest:
9.2×1010, 6.4×1015, 2.1×1020, 1.7×1015
5.63×10-5, 4.16×10-3, 3.42×10-6, 8.71×10-3