Transcript Scientific Notation

Warm up…
• Simplify
1.
3 5
s t
( s2 t 3 ) 1
2. The number of three-letter “words” that can be
formed with the English alphabet is 263. the
number of five-letter “words” that can be formed
265. How many times more five-letter “words” can
be formed that three-letter words?
© William James Calhoun, 2001
Scientific Notation
© William James Calhoun, 2001
OBJECTIVES:
• You will simplify expressions involving
quotients of monomials and simplify
expressions containing negative exponents.
© William James Calhoun, 2001
In science, very small numbers are found when measuring energy levels
of electrons and atomic radii.
Also in science, extremely large numbers are found when measuring
distances between stars or energy levels of nuclear explosions.
Extremely small numbers have lots of decimal places with most of those
decimals being zeros.
Extremely large numbers have lots of zero place holders.
People working with such numbers wanted to find an easier way to write
these numbers without having to put in all those zeros.
© William James Calhoun, 2001
Enter: Scientific Notation.
Scientific Notation is a way to write either very large or very small
numbers without all the zeros needed for place-holding.
It can also be used to write any numbers.
To write a number in scientific notation, you must first get the decimal
place behind the very first digit of the number.
Count how many places you moved the decimal.
Add “x10” to the end of your number and raise the “x10” to a power
based on the number of places the decimal moved.
The “x10power” alleviates the zeros.
© William James Calhoun, 2001
For example, there are 29,000,000 passengers that arrive and depart from
Miami International Airport.
That number has a lot of zeros. Scientific Notation is helpful in this case
to write 29 million in a more compact form.
The decimal needs to be behind the “2” - the first digit of the number.
It is currently behind the last zero.
29,000,000.
.
The decimal moved seven places to the left.
Written in scientific notation, 29 million is 2.9 x107.
DEFINITION OF SCIENTIFIC NOTATION
A number is expressed in scientific notation when it is in the form
a x 10n,
where 1  a < 10 and n is an integer.
© William James Calhoun, 2001
With very small numbers in scientific notation, the exponent on the x10
will be a negative number.
Knowing whether to make the exponent on the x10 positive or negative
is the hardest part of scientific notation for most students.
You need to remember:
Positive x10 means the number is bigger written out the long way.
So there usually needs to be some zeros as place holders at the end of the number.
8.3 x105 = 830,000
Negative x10 means the number is smaller written out the long way.
So there usually needs to be some zeros as place holders between the decimal and the
first non-zero number.
8.3 x10-5 = 0.000083
© William James Calhoun, 2001
EXAMPLE 1: Express each number in scientific notation.
A. 98,700,000,000
B. 0.0000056
Move the decimal. Remember the decimal is
hidden at the end.
The decimal moved how many places?
10
The number is larger than its scientific
notation, so the exponent will be…
positive.
The answer is then:
Move the decimal.
The decimal moved how many places?
6
The number is larger than its scientific
notation, so the exponent will be…
negative.
The answer is then:
9.8 x1010
5.6 x10-6
The book shows a different way of doing this example. Use that method
if you wish.
I prefer my way of doing scientific notation.
© William James Calhoun, 2001
You try…
•
Express each number in scientific
notation.
1. 915,600,000,000
2. 0.003157
© William James Calhoun, 2001
EXAMPLE 2: Express each number in standard notation.
A. 3.54 x105
B. 9.72 x10-4
The power is positive, so the number in
standard should be larger than in scientific
notation.
To make a larger number, we need to move
the decimal to the…
right.
How many places?
5
Do so.
3.54000
Insert place holders between the decimal and
the next nearest digit.
354,000
The power is negative, so the number in
standard should be smaller than in scientific
notation.
To make a smaller number, we need to move
the decimal to the…
left.
How many places?
4
Do so.
0009.72
Insert place holders between the decimal and
the next nearest digit.
Re-write putting a zero in front to notice the
decimal point.
0.000972
© William James Calhoun, 2001
EXAMPLE 3: Use scientific notation to evaluate each expression.
A. (610)(2,500,000,000)
B. (0.000009)(3700)
Some calculators can handle really big/small numbers. Others cannot. For the others that cannot, there is
a way to use scientific notation to get answers.
Also, most calculators nowadays can input scientific notation using the “x10” key or the “ EE” key.
First, we will rewrite both problems in scientific notation.
A. (6.1 x102)(2.5 x109)
A. (9 x10-6)(3.7 x103)
Now, the “x10” is the same base in both parts. Also, we are dealing with multiplication. When you
multiply a same base, you simply…
add the exponents.
So, multiply the number parts together then add the exponents on the x10.
(6.1 x 2.5)(102 x 109)
15.25 x1011
(9 x 3.7)(10-6 x 103)
33.3 x10-3
Get these two answers back into scientific notation by move the decimal over on place to the left.
1.525 x1012
3.33 x10-2
= 1,525,000,000,000
= 0.0333
© William James Calhoun, 2001
EXAMPLE 4: Use scientific notation to evaluate the division
expression:
2.0286x108
3.15x103
First, divide the number in the top by the number in the bottom, ignoring the x10 notation.
2.0286
3.15
 0.644
Take care of the x10 notation.
x108
(8 - 3) = x105
3 = x10
x10
Rewrite, then change the answer so the decimal is behind the first digit.
0.644
. x105 = 6.44 x104
© William James Calhoun, 2001
You try…
• Evaluate. Express each result in scientific
and standard notation.
3
.  10
1. (5  102 )(2.3  1012 )
2. 312
156
.  10
3
© William James Calhoun, 2001
CLASS WORK
Page 428
#16-17, 18-57 multiples of 3
And number 58
© William James Calhoun, 2001