Unit 7 Exponents And Scientific Notation — Unit Plan

Title

Assessment

Lesson 1

Exponent Review

Exponent Check

What is the value of $3^4$ ?
How many times bigger is $3^{15}$ compared to $3^{12}$ ?

Show Solution

81, because $3^4 = 3\boldcdot 3 \boldcdot 3 \boldcdot 3 = 9 \boldcdot 3 \boldcdot 3 = 27 \boldcdot 3 = 81$ .
$3^{15}$ is 27 times larger than $3^{12}$ , because $3^{15}$ has 3 more factors that are 3 and $3^3 = 27$ .

Lesson 2

Multiplying Powers of 10

That's a Lot of Office Space!

Rewrite $10^{32} \boldcdot 10^{6}$ using a single exponent.
A company leases out office space for $10^2$ dollars per square foot. If this company owns approximately $10^6$ square feet of office space in multiple locations worldwide, how much money could they make renting out all of their office space? Express your answer both as a power of 10 and as a dollar amount.

Show Solution

$10^{38}$ , because $10^{32} \boldcdot 10^{6} = 10^{32+6} = 10^{38}$ .
$10^8$ or $100,000,000.

Lesson 3

Powers of Powers of 10

Making a Million

Here are some equivalent ways of writing $10^4$ :

10,000
$10 \boldcdot 10^3$
$(10^2)^2$

Write as many expressions as you can that have the same value as $10^6$ .

Show Solution

Answers vary. Sample responses:

1,000,000
$1,000\boldcdot 1,000$
$10^2 \boldcdot 10^4$
$(10^3)^2$
$1\boldcdot 10^6$
$10\boldcdot10\boldcdot10\boldcdot10\boldcdot10\boldcdot10$

Lesson 4

Dividing Powers of 10

Why Subtract?

Why is $\dfrac{10^{15}}{10^4}$ equal to $10^{11}$ ? Explain or show your thinking.

Show Solution

Sample response: $\frac{10^{15}}{10^4} = 10^{11}$ because 4 factors that are 10 in the numerator and denominator are used to make 1, leaving 11 remaining factors that are 10. In other words, $\displaystyle \frac{10^{15}}{10^4} = \frac{10^4 \boldcdot 10^{11}}{10^4}=10^{11}.$

Lesson 5

Negative Exponents with Powers of 10

Negative Exponent True or False

Mark each of the following equations as true or false. Explain or show your reasoning.

$10^{\text -5} = \text -10^5$
$(10^2)^{\text -3} = (10^{\text -2})^3$
$\dfrac{10^3}{10^{14}} = 10^{\text-11}$

Show Solution

False. Sample reasoning: $10^{\text-5} = \frac{1}{100,000}$ , whereas $\text-10^5 = \text-100,000$ .
True. Sample reasoning: Both $\left(10^2 \right)^{\text-3}$ and $\left(10^{\text-2} \right)^{3}$ are equal to $10^{\text-6}$ .
True. Sample reasoning: $\frac{10^3}{10^{14}} = 10^{3-14} = 10^{\text-11}$ .

Section A Check

Section A Checkpoint

Problem 1

Select all the expressions that are equivalent to

10^{\text-3}

A.1,000

\frac1{1,000}

\frac1{10^3}

\frac{1}{10^{\text-3}}

\text-\frac1{1,000}

F.-1,000

(\frac1{10})^3

Show Solution

B, C, G

Problem 2

Write each expression using a single exponent.

$\dfrac{10^5\boldcdot(10^4)^3}{10^7}$
$\dfrac{10^0\boldcdot10^1\boldcdot10^{11}}{(10^2)^3}$

Show Solution

$10^{10}$
$10^6$

Lesson 6

What about Other Bases?

Spot the Mistake

Diego was trying to write $2^3 \boldcdot 2^2$ with a single exponent and wrote $2^3 \boldcdot 2^2 = 2^{3 \boldcdot 2} = 2^6$ . Do you agree with Diego? Explain your reasoning.
Andre was trying to write $\dfrac{7^4}{7^{\text -3}}$ with a single exponent and wrote $\dfrac{7^4}{7^{\text -3}} = 7^{4-3} = 7^1$ . Do you agree with Andre? Explain your reasoning.

Show Solution

I do not agree with Diego. Sample reasoning: Diego multiplied the exponents when he should have added them. To see this, he could have expanded the expressions: $2^3 \boldcdot 2^2 = (2 \boldcdot 2 \boldcdot 2)(2 \boldcdot 2) = 2^{3 +2} = 2^5$ .
I do not agree with Andre. Sample reasoning: Andre did $7^{4-3}$ when he should have done $7^{4-(\text- 3)}$ to get $7^7$ .

Lesson 7

Practice with Rational Bases

Working with Exponents

Rewrite each expression using a single, positive exponent:
1. $\dfrac{9^3}{9^9}$
2. $14^{\text-3} \boldcdot 14^{12}$
Diego wrote $6^4 \boldcdot 8^3 = 48^7$ . Explain what Diego’s mistake was and how you know the equation is not true.

Show Solution

1. $\frac{1}{9^6}$
2. $14^9$
Sample response: Diego multiplied the bases and added their exponents. The equation is not true because 4 repeated factors that are 6 multiplied by 3 repeated factors that are 8 is much smaller than 7 repeated factors that are 48.

Lesson 8

Combining Bases

Equivalent or Not?

Determine whether each of these expressions is equivalent or not equivalent to $(12\boldcdot 4)^7$ . Explain or show your reasoning.

$12^7 \boldcdot 4^7$
$12 \boldcdot 4^7$
$(12\boldcdot 7)\boldcdot (4\boldcdot 7)$
$48^7$

Show Solution

Equivalent. Sample reasoning: Since there are 7 factors that are 12 and 7 factors that are 4, they can be regrouped as 7 factors of $(12\boldcdot 4)$ .
Not equivalent. Sample reasoning: This expression has only 1 factor that is 12 instead of 7 factors that are 12.
Not equivalent. Sample reasoning: In this expression 12 and 4 are each being multiplied by 7 instead of being raised to the power of 7.
Equivalent. Sample reasoning: Since $12\boldcdot 4=48$ these expressions are equivalent.

Section B Check

Section B Checkpoint

Problem 1

Select all the expressions that are equivalent to

2^{5}

2\boldcdot2\boldcdot2\boldcdot2\boldcdot2

5\boldcdot5

\frac{10^{10}}{5^2}

\frac{10^{10}}{5^5}

2^{\text-2}\boldcdot2^7

\frac{2^3\boldcdot2^5}{2^3}

\text-2^{\text-5}

2^0\boldcdot2^5

Show Solution

A, E, F, H

Problem 2

Write

4^5\boldcdot7^5

using a single exponent.

Show Solution

28^5

Problem 3

Explain why

3^4\boldcdot8^3

cannot be written using a single exponent.

Show Solution

Sample response: The expression has 4 factors that are 3 and 3 factors that are 8. Three 3s and three 8s can be regrouped into 3 factors of

3\boldcdot8

, but there will still be 1 extra factor of 3 that cannot be regrouped and, so, this expression cannot be rewritten using a single exponent.

Lesson 9

Describing Large and Small Numbers Using Powers of 10

Better with Powers of 10

Write 0.000000123 as a multiple of a power of 10.
Write 123,000,000 as a multiple of a power of 10.

Show Solution

Sample response: $(1.23) \boldcdot 10^\text{-7}$ (or equivalent)
Sample response: $(1.23) \boldcdot 10^8$ (or equivalent)

Lesson 10

Representing Large Numbers on the Number Line

Describe the Point

The speed of light through ice can be written as a multiple of a power of 10, such as $(2.3) \boldcdot 10^8$ meters per second, or as a value, such as 230,000,000 meters per second. Use the number line to answer questions about points $A$ and $B$ .

A number line, 11 tick marks, 0, blank, B, 7 blank tick marks, 10 to the ninth power. The sixth and seventh tick marks are zoomed out to 11 tick marks and the fourth is labeled A.

Describe point $A$ as:
1. A multiple of a power of 10
2. A value
Describe point $B$ as:
1. A multiple of a power of 10
2. A value
Plot a point $C$ that is greater than $A$ and less than $B$ . Describe point $C$ as:
1. A multiple of a power of 10
2. A value

Show Solution

1. $2 \boldcdot 10^8$ .
2. 200,000,000.
1. $(5.3) \boldcdot 10^8$ .
2. 530,000,000.
Sample response: any value between $2 \boldcdot 10^8$ and $(5.3) \boldcdot 10^8$ .

Lesson 11

Representing Small Numbers on the Number Line

Describing Very Small Numbers

Write 0.00034 as a multiple of a power of 10
Write $(5.64) \boldcdot 10^{\text -7}$ as a decimal.

Show Solution

$(3.4) \boldcdot 10^{\text-4}$ , $34 \boldcdot 10^{\text-5}$ , or equivalent
0.000000564

Lesson 12

Applications of Arithmetic with Powers of 10

That's a Lot of Cells

There are about 260 million adults in the United States and the average adult has 10 pints of blood. If there are $2.4\times10^{12}$ red blood cells in one pint of blood, about how many red blood cells are there in all the adults in the United States?

Show Solution

There are $6.24\times10^{21}$ red blood cells since $(2.6\times10^8)\boldcdot(10)\boldcdot(2.4\times10^{12})=6.24\times10^{21}$ .

Section C Check

Section C Checkpoint

Problem 1

For each pair, determine which quantity is larger, and estimate how many times larger.

1. The country of Canada covers approximately 10 million square kilometers.
2. The country of Mexico covers approximately $2\boldcdot10^6$ square kilometers.
1. The diameter of one atom of hydrogen is $5\boldcdot10^{\text-11}$ meters.
2. The diameter of one atom of magnesium is $3\boldcdot10^{\text-10}$ meters.

Show Solution

The country of Canada covers an area about 5 times the size of the country of Mexico, because $10,000,000=10\boldcdot10^6$ , which is 5 times larger than $2\boldcdot10^6$ .
The diameter of one atom of magnesium is 6 times larger than the diameter of one atom of hydrogen, because $3\boldcdot10^{\text-10}=30\boldcdot10^{\text-11}$ , which is 6 times larger than $5\boldcdot10^{\text-11}$

Problem 2

Fingernails grow at a rate of about

10^{\text-9}

meters per second. How long would it take a fingernail to grow 1 centimeter? Explain or show your reasoning.
(There are 100 centimeters in 1 meter.)

Show Solution

10^7

or 10,000,000 seconds (about 166,667 minutes or 2,778 hours or 115 days or almost 4 months). Sample reasoning: Since a fingernail grows

10^{\text-9}

meter per second, it will take

10^9

seconds to grow 1 meter. Since there are 100 or

10^2

centimeters in 1 meter,

\frac{10^9}{10^2}=10^7

Lesson 14

Estimating with Scientific Notation

Estimate how many times larger $6.1 \times 10^7$ is than $2.1 \times 10^{\text -4}$ . Explain or show your reasoning.
Estimate how many times larger $1.9 \times 10^{\text -8}$ is than $4.2 \times 10^{\text -13}$ . Explain or show your reasoning.

Show Solution

$6.1 \times 10^7$ is about 300 billion times larger than $2.1 \times 10^{\text-4}$ . Sample reasoning: $\displaystyle \frac{6.1 \times 10^7}{2.1 \times 10^{\text-4}} \approx \frac{6 \times 10^7}{2 \times 10^{\text-4}} = 3 \times 10^{7 - (\text-4)} = 3 \times 10^{11}.$
$1.9 \times 10^{\text-8}$ is about 50,000 times larger than $4.2 \times 10^{\text-13}$ . Sample reasoning: $\displaystyle \frac{1.9 \times 10^{\text-8}}{4.2 \times 10^{\text-13}} \approx \frac{2 \times 10^{\text-8}}{4 \times 10^{\text-13}} = 0.5 \times 10^5 = 5 \times 10^4.$

Section D Check

Section D Checkpoint

Problem 1

Select all expressions written using scientific notation:

A.42,000,000

56\times1,000,000

87\times10^6

4.2\times10^7

5\times10^{\text-3}

0.2\times10^9

G.1.4 million

0.009\times10^{10}

6\times10\times10\times10

7.3E4

Show Solution

D, E, J

Problem 2

The table shows the average mass of one individual creature and an estimated total number of those creatures on Earth.

creature	total number	mass of one individual (kg)
humans	$7.5 \times 10^9$	$6.2 \times 10^1$
cows	$1.3 \times 10^9$	$4 \times 10^2$
sheep	$1.75 \times 10^9$	$6 \times 10^1$
chickens	$2.4 \times 10^{10}$	$2 \times 10^0$
ants	$5 \times 10^{16}$	$3 \times 10^{\text -6}$
blue whales	$4.7 \times 10^3$	$1.9 \times 10^5$
antarctic krill	$7.8 \times 10^{14}$	$4.86 \times 10^{\text -4}$
zooplankton	$1 \times 10^{20}$	$5 \times 10^{\text -8}$
bacteria	$5 \times 10^{30}$	$1 \times 10^{\text -12}$

What is the total mass of 1 human, 1 cow, and 2 chickens? Explain or show your reasoning.
What is the total mass of all the zooplankton on the planet? Explain or show your reasoning.

Show Solution

466 kg. Sample reasoning: Each human is 62 kg, each cow is 400 kg, and each chicken is 2 kg. The total mass for 1 human, 1 cow and 2 chickens would be $62+400+2+2=466$ kg.
$5\times10^{12}$ kg (or equivalent). Sample reasoning: $(1\times10^{20})\times(5\times10^{\text-8})=(1\times5)\times(10^{20}\times10^{\text-8})=5\times10^{12}$ .

Lesson 16

Is a Smartphone Smart Enough to Go to the Moon?

No cool-down

Unit 7 Assessment

End-of-Unit Assessment

Problem 1

Select all the expressions that equal $4 \times 10^6$ .

$(2 \times 10^8)(2 \times 10^{\text-2})$

$40 \times 10^5$

$40^6$

400,000

$\dfrac{1.2 \times 10^9}{3 \times 10^2}$

Show Solution

A, B, E

Problem 2

Select all the expressions that equal $6^{\text-10}$ .

$6^{\text-5} \boldcdot 6^2$

$\frac 1 {6^{10}}$

$\left(6^{\text-5}\right)^2$

$\dfrac{6^{\text-3}}{6^{7}}$

$\dfrac{6^5 \boldcdot 6^{\text-3}}{6^{\text-8}}$

Show Solution

B, C, D

Problem 3

About $3.9 \times 10^7$ people live in California. About $1.3 \times 10^6$ people live in Maine. About how many more people live in California than live in Maine?

$2.6 \times 10^6$

$2.6 \times 10^7$

$3.77 \times 10^6$

$3.77 \times 10^7$

Show Solution

$3.77 \times 10^7$

Problem 4

What number is represented by point $P$ ?

Show Solution

$5.7 \times 10^{\text-3}$ or $0.0057$ (or equivalent) (Point $P$ lies between $5 \times 10^{\text-3}$ and $6 \times 10^{\text-3}$ . In the zoomed-in number line, point $P$ lies on the seventh of ten tick marks, so it represents $5.7 \times 10^{\text-3}$ .)

Problem 5

In 2015, there were about 22 million teenagers (aged 13–17) in the United States. They each sent an average of 900 text messages per month. Estimate how many text messages were sent in 2015 by teenagers in the United States in one month. Express your answer using scientific notation.

Show Solution

An estimate between $1.8 \times 10^{10}$ and $2.2 \times 10^{10}$ is reasonable. (Multiply $(22 \times 10^6) \boldcdot (9 \times 10^2) = 198 \times 10^8$ . Then express in scientific notation as $1.98 \times 10^{10}$ . Students may also use $20\times10^6$ as an estimate for 22 million, or 1,000 as an estimate for 900 texts.)

Problem 6

Place a number in each box so that each equation is true.

$\large 2^{\boxed{\phantom{0}}} \boldcdot 2^{\boxed{\phantom{0}}} = 2^0$
$\large \dfrac{2^3}{2^{\boxed{\phantom{0}}}} = 2^{\boxed{\phantom{0}}}$
$\large 2^{\text -3} \boldcdot {\boxed{\phantom{0}}}^{\text{ -}3} = 10^{\boxed{\phantom{0}}}$

Show Solution

Answers vary. Sample response: $2^5 \boldcdot 2^{\text-5} = 2^0$
Answers vary. Sample response: $\frac{2^3}{2^5} = 2^{\text-2}$
$2^{\text-3} \boldcdot 5^{\text-3} = 10^{\text-3}$

Minimal Tier 1 response:

Work is complete and correct.

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: One of the three equations produced is incomplete or incorrect.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Two or more of the three equations produced are incomplete or incorrect.

Problem 7

Here are the approximate populations of three countries, expressed in scientific notation: Panama: $4 \times 10^6$ ; Peru: $3.2 \times 10^7$ ; Thailand: $7 \times 10^7$ .

Lin says that more than 20 times as many people live in Thailand than in Panama. Do you agree with Lin? Explain or show your reasoning.
What is the population of all three countries together?

Show Solution

I do not agree with Lin. Sample reasoning: If more than 20 times as many people lived in Thailand as in Panama, the population of Thailand would be at least 80 million, but it is only 70 million.
$1.06\times10^8$ people or about 100 million people (or equivalent)

Minimal Tier 1 response:

Work is complete and correct, with complete explanation or justification.
Acceptable errors: Estimates that are reasonable but do not match solution exactly

Tier 2 response:

Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
Sample errors: Statement agreeing or disagreeing with Lin is missing, though explanation of reasoning implies agreement; invalid or incorrect explanation for why there aren’t more than 20 times as many people in Thailand; minor calculation error when finding sum of the 3 countries’ populations, but correct interpretation of place value.

Tier 3 response:

Work shows a developing but incomplete conceptual understanding, with significant errors.
Sample errors: Two or more error types from Tier 2 response; incorrectly stating there are more than 20 times as many people in Thailand; failing to find equivalent expressions with the same place value before adding or subtracting; incorrectly interpreting numbers written in scientific notation.

Tier 4 response:

Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery.
Sample errors: Two or more error types from Tier 3 response.