(Positive & Negative) Exponents
8.EE.3 Use number expressed in the form of a single digit times an integer power of
10 to estimate very large or very small quantities, and to express how many
times as much one is that the other.
8.EE.4 Perform operations with numbers expressed in scientific notation, including
problems where both decimal and scientific notation are used. Use scientific
notation and choose units of appropriate size for measurements of very large
or very small quantities.
 Students know that positive powers of 10 are very large numbers, and negative
powers of 10 are very small numbers.
 Students know that the exponent of an expression provides information about the
magnitude of a numbers.
 Students compare and estimate quantities in the form of a single digit times a
power of 10.
 Students use their knowledge of ratios, fractions, and laws of exponents to simplify
expressions.
https://www.youtube.com/watch?v=GKQdkS0qG3g
Despicable Me - Vector
WKSP
HW
Procedure
Media
Student Outcomes
Standard
Pre Requisite
Magnitude
#16
Spiral Review #4
Magnitude
Compare the following:
105
1050
105
10-5
10-5
10-50
Powers of 10
Let M represent the world population as of March 23, 2013. (M ≈ 7,073,981,143)
How many digits does this number have?
Compare: M
1010
We say that the 10th power of 10 exceeds M.
Let M represent the US national debt on March 23, 2013. (M ≈ 16,755,133,009,522 to
the nearest dollar)
What is the smallest power of 10 that can exceed M?
Exercise: Find the smallest power of 10 that will exceed M.
1. Let M = 993,456,789,098,765
2.
899
Let M = 78,491 987
What kind of exponent represents a very small number?
The average ant weighs 0.0003 grams.
3
0.0003 =
= 3 × 10
0
1
The mass of a neutron is 0.000 000 000 000 000 000 000 000 001 674 9 kilograms.
We would need to swing the decimal 27 place values before a digit appears!
What power of 10 can be used to simplify the mass of a neutron? 10
Exercise: Express each small number as a negative power of 10.
1. The chance of having the same DNA as another person is approximately 1 in 10
trillion.
1
1
=
= 10
10,000,000,000,000
10
3.
The chance of winning a big lottery prize is about 10-8, and the chance of being
struck by lightning in the US in any given year is about 0.000001. Which do you
have a greater chance of experiencing? Explain.
Magnitude
Estimating Quantities
Example: In 1723, the population of NY City was about 7,248. By 1870, almost 150
years later, the population had grown to 942,292. Approximately, how many
times greater was the population in 1870 compared to 1723?
‘Approximately’ signals us to use powers of 10. Thank goodness!
Population in 1723 ≈ 10
Population in 1870 ≈ 10
‘Times’ signals that multiplication was exercised to get to the population in 1870. We’re
reversing the process. What operation should we use?
Running theme in math (
𝑓𝑖𝑛𝑖𝑠ℎ
𝑠𝑡𝑎𝑟𝑡
) will help us finish the problem.
Exercise:
1. The Federal Reserve states that the average household in January of 2013 had
$7,122 in credit card debt. About how many times greater is the US national debt,
which is $16,755,133,009,522?
2.
About how many times greater is the population of NY State (19,570,261)
compared to that of NY City (8,336,697)?
Precision
If we want to get more precise, we can go down a place value.
NY State: 19,570,261 rounded to the nearest million is _________________ = 2 x 10 ?
NY City: 8,336,697 rounded to the nearest million is __________________ = 8 x 10 ?
We now have: ____________
Exercise:
1. There are about 9 billion devices connected to the internet. If a wireless router can
support 300 devices, how many wireless routers are necessary to connect all 9
billion devices wirelessly?
2.
The average American household spends about $40,000 each year. If there are
about 1 x 108 households, what is the total amount of money spent by American
households in one year?
Magnitude
Name: __________________________________
Pre-Algebra
Date: ______
Exit Ticket
1.
Let M = 118,526.65902. Find the smallest power of 10 that will exceed M.
2.
The average person takes about 30,000 breaths per day. Express this number as a
single-digit integer times a power of 10.
If the average American lives about 80 years (or about 30,000 days), how many
total breaths will a person take in her lifetime?
Magnitude
Name: _______________________________________
Pre-Algebra
Date: _____
HW #16
1. Place each of the following numbers on a number line in its approximate location.
10-99
10-17
1014
10-5
1030
Express the following using positive exponents. Then represent as a rational number.
2. What is the smallest power of 10 that would exceed 987,654,321,098,765,432?
3. Which number is equivalent to 0.0000001? 107 or 10-7? How do you know?
4. Sarah said that 0.00001 is bigger than 0.001 because the first number has more
digits to the right of the decimal point. Is Sarah correct? Explain your thinking
using negative powers of 10 and the number line.
5. There are about 100 million smartphones in the US. Your friend has a smartphone.
What share of US smartphones does your friend have?
1
1
=
= 10
100,000,000
10
6. There are about 3,000,000 students attending school, kindergarten through 12 th
grade, in New York. Express the number of students as a single-digit integer times a
power of 10.
The average number of students attending a middle school in New York is 8 x 10 2.
How many times greater is the overall number of k-12 students compared to the
number of middle school students?
Magnitude
Review:
7. Divide the following:
a. 94.24 ÷ 4 =
c.
18
−2
=
8. Simplify the following:
1
1
a. 8 + 4 =
c.
1
8
∙
b. 24.68 ÷ 0.2 =
1
4
=
b.
d.
1
8
1
8
−
÷
1
4
1
4
=
=