Grade 8 Module 1 MMA Review Answer Key
1. The number of users of social media has increased significantly since the year 2001. In fact, the
approximate number of users has quadrupled each year. It was reported that in 2005 there were 4
million users of social media.
a.
Assuming that the number of users continues to quadruple each year, for the next three years,
determine the number of users in 2006, 2007, and 2008.
2005 – 4 million users
2006 – 16 million users
2007 – 64 million users
2008 – 256 million users
b.
c.
Assume the trend in the numbers of users quadrupling each year was true for all years from
2001 to 2009. Complete the table below using 2005 as year 1 with 4 million as the number of
users that year.
Year
-3
-2
-1
0
1
2
3
4
5
# of
users in
millions
𝟏
𝟔𝟒
𝟏
𝟏𝟔
𝟏
𝟒
1
4
16
64
256
1024
Given only the number of users in 2005 and the assumption that the number of users
quadruples each year, how did you determine the number of users for years 2, 3, 4, and 5?
The number of users quadrupling means that the number of users is multiplied by 4 every year.
d.
Given only the number of users in 2005 and the assumption that the number of users
quadruples each year, how did you determine the number of users for years 0, -1, -2, and -3?
When we quadruple, we multiply by 4 so if we go backwards, then we would divide by 4.
e.
Write an equation to represent the number of users in millions, 𝑁, for year, t,
t ³ -3.
𝑵 = 𝟒𝒕
f.
Using the context of the problem, explain whether or not the formula, 𝑁 = 4𝑡 would work for
finding the number of users in millions in year 𝑡, for all t ≤ 0.
We only know that the number of users quadrupled from the year 2001 and on. We do not
know if the number of users quadruples before 2001 so we cannot predict the number of
users for 2000 and the years before that.
g.
Assume the total number of users continues to quadruple each year after 2009. Determine the
number of users in 2011. Given that the world population at the end of 2010 was
approximately 7 billion, is this assumption reasonable? Explain your reasoning.
2005 – 4,000,000 users
2006 – 16,000,000 users
2007 – 64,000,000 users
2008 – 256,000,000 users
2009 – 1,024,000,000 users
2010 – 4,096,000,000 users
2011 – 16,384,000,000 users
In the year 2011, there would be 16 billion users. This is not reasonable because the population in
2010 was only approximately 7 billion so it is very unlikely that 16 billion people would even be
alive.
2. Let 𝑚 be a whole number.
a.
Use the properties of exponents to write an equivalent expression that is a product of unique
primes, each raised to an integer power.
620 ∙ 105
305
(𝟐 ∙ 𝟑)𝟐𝟎 ∙ 𝟏𝟎𝟓
(𝟑 ∙ 𝟏𝟎)𝟓
𝟐𝟐𝟎 ∙ 𝟑𝟐𝟎 ∙ 𝟏𝟎𝟓
𝟑𝟓 ∙ 𝟏𝟎𝟓
𝟐𝟐𝟎 ∙ 𝟑𝟏𝟓
b.
Use the properties of exponents to prove the following identity:
64𝑚 ∙ 10𝑚
= 24𝑚 ∙ 32𝑚
30𝑚
(𝟐 ∙ 𝟑)𝟒𝒎 ∙ 𝟏𝟎𝒎
(𝟑 ∙ 𝟏𝟎)𝒎
𝟐𝟒𝒎 ∙ 𝟑𝟒𝒎 ∙ 𝟏𝟎𝒎
𝟑𝒎 ∙ 𝟏𝟎𝒎
𝟐𝟒𝒎 ∙ 𝟑𝟑𝒎
c.
What value of 𝑚 could be substituted into the identity in part (b) to find the answer to part (a)?
m = 5 then 𝟐𝟒𝒎 ∙ 𝟑𝟑𝒎 = 𝟐𝟐𝟎 ∙ 𝟑𝟏𝟓
3.
a.
Jill writes 32 ∙ 92 = 274 and the teacher marked it wrong. Explain Jill’s error.
You cannot add exponents with different bases. You can only add exponents when two
expressions have the same base. You also do not multiply the base numbers either. You
would keep the base the same and just add the exponents.
b.
Find 𝑛 so that the number sentence below is true:
32 ∙ 92 = 32 ∙ 3𝑛 = 36
𝟑𝟐 ∙ 𝟗𝟐
𝟑𝟐 ∙ (𝟑 ∙ 𝟑)𝟐
𝟑𝟐 ∙ 𝟑𝟐 ∙ 𝟑𝟐
𝟑𝟐 ∙ 𝟑𝟒
𝒏=𝟒
c.
Use the definition of exponential notation to demonstrate why 32 ∙ 92 = 36 is true.
𝟑𝟐 ∙ 𝟗𝟐
𝟑𝟐 ∙ (𝟑 ∙ 𝟑)𝟐
𝟑𝟐 ∙ 𝟑𝟐 ∙ 𝟑𝟐
𝟑𝟔
d.
You write 63 ∙ 6−8 = 6−5. Keisha challenges you, “Prove it!” Show directly why your answer is
correct without referencing the Laws of Exponents for integers, i.e., 𝑥 𝑎 ∙ 𝑥 𝑏 = 𝑥 𝑎+𝑏 for positive
numbers 𝑥 and integers 𝑎 and 𝑏.
𝟔𝟑
𝟔𝟖
𝟔𝟑
𝟔𝟑 ∙ 𝟔𝟓
𝟏
𝟔𝟓
𝟔−𝟓