8.1 Lesson 9 8.2.notebook
Lesson 9
Scientific Notation
Student Outcome: Students write, add, and subtract numbers in scientific notation and understand what is meant by the term leading digit.
8.1 Lesson 9 8.2.notebook
IMPORTANT!
A positive, finite decimal s is said to be written in scientific notation if it is expressed as a product d x 10n, where d is a finite decimal so that 1 < d < 10, and n is an integer.
The integer n is called the order of magnitude of the decimal d x 10n.
8.1 Lesson 9 8.2.notebook
Let's Take A Look
The finite decimal 234.567 is equal to every one of the following:
2.34567 x 102
0.234567 x 103
23.4567 x 10
234.567 x 100
234567 x 10­3
234567000 x 10­6
8.1 Lesson 9 8.2.notebook
On Your Own
Are the following numbers written in scientific notation? If no, state the reason.
Exercise 1
1.908 x 1017
Exercise 4
yes
Exercise 2
0.325 x 10­2
no, it must be a product
Exercise 5
no, d < 1
Exercise 3
7.99 x 1032
4.0701 + 107
18.432 x 58
no, d > 10 and it is x 5
Exercise 6
yes
8 x 10­11
yes
8.1 Lesson 9 8.2.notebook
On The Numberline
Exponent n is called the order of magnitude of the positive number s = d x 10n (in scientific notation ) because the following inequalities hold:
10n < s and s < 10n+1
The exponent n serves to give an approximate location of s on the number line. n gives the approximate magnitude of s.
The number s is between 10n and 10n+1.
8.1 Lesson 9 8.2.notebook
Example 2
Step by Step Strategies
Let's say we need to determine the difference in the populations of Texas and North Dakota. In 2012, Texas had a population of about 26 million people, and North Dakota had a population of about 6.9 x 104.
Texas: 26,000,000 = 2.6 x 107. North Dakota: 69,000 = 6.9 x 104.
To ind the difference we subtract:
2.6 x 107 ‑ 6.9 x 104 In order to add or subtract, we need the same order of magnitude and same base. By order of magnitude, we can think "raised to the same exponent." We can use the distributive property to perform operations because each number has a common factor. Speciically for this problem, we can rewrite the population of Texas so that it is an integer multiplied by 104, and then subtract.
2.6 x 107 ‑ 6.9 x 104 = 2.6 x 103 x 104 ‑ 6.9 x 104
Remember, using laws of exponents, when multiplying the same bases, we add the exponents. Therefore, we can rewrite 107 as 103 x 104.
= 2600 x 104 ‑ 6.9 x 104
When we multiply 2.6 x 103 we get 2600.
Now we have matching factors in the form of 104. Remember, factors have to be raised to the same exponent in order to add or subtract.
= 2600 x 104 ‑ 6.9 x 104
= (2600 ‑ 6.9) x 104
Since both factors have 104 in common, we can rewrite using the Distributive Property.
It may be more familiar in the form: 104(2600 ‑ 6.9). It is the same thing, just written differently because we are working with scientiic notation.
= 2593.1 x 104
Is this in the form of scientiic notation? (The factor, 2593.1, needs to be less than 10 and greater than or equal to 1.)
= 2.5931 x 107 Scientiic Notation form
or
= 25,931,000 Standard form
8.1 Lesson 9 8.2.notebook
Example 3
H20
We can use scientiic notation to help us to ind the combined mass of two hydrogen atoms and one oxygen atom, otherwise know as Water!
8.1 Lesson 9 8.2.notebook
Exercises 7 and 8
Use the table below to complete Exercises 7 and 8. The table shows the debt of the three most populous states and the three least populous states.
Exercise 7
a. What is the sum of the debts for the three most populous states?
Express your answer in scientific notation.
b. What is the sum of the debt for the three least populous states?
Express your answer in scientific notation.
c. How much larger is the combined debt of the three most populous states than that of the three least populous states? Express your answer in scientific notation.
8.1 Lesson 9 8.2.notebook
Exercises 7 and 8
Exercise 8
a. What is the sum of the population for the three most populous states?
Express your answer in scientific notation.
b. What is the sum of the population for the three least populous states?
Express your answer in scientific notation.
c. Approximately how many times greater is the total population of California, New York and Texas compared to the total population of North Dakota, Vermont, and Wyoming?
8.1 Lesson 9 8.2.notebook
Exercise 9
Al1 planets revolve around the sun in elliptical orbits. Uranus's furthest distance from the sun is approximately 3.004 x 109 km, and its closest distance is approximately 2.749 x 109 km. Using this information, what is the average distance of Uranus from the sun?
8.1 Lesson 9 8.2.notebook
Problem Set