Unit 7 Exponents And Scientific Notation — Unit Plan

TitleTakeawaysVisual / Anchor ChartAssessment
Lesson 1
Exponent Review
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Exponent Check (1 problem)
  1. What is the value of 343^4?
  2. How many times bigger is 3153^{15} compared to 3123^{12}?
Show Solution
  1. 81, because 34=3⋅3⋅3⋅3=9⋅3⋅3=27⋅3=813^4 = 3\boldcdot 3 \boldcdot 3 \boldcdot 3 = 9 \boldcdot 3 \boldcdot 3 = 27 \boldcdot 3 = 81.
  2. 3153^{15} is 27 times larger than 3123^{12}, because 3153^{15} has 3 more factors that are 3 and 33=273^3 = 27.
Lesson 2
Multiplying Powers of 10
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That's a Lot of Office Space! (1 problem)
  1. Rewrite 1032⋅10610^{32} \boldcdot 10^{6} using a single exponent.
  2. A company leases out office space for 10210^2 dollars per square foot. If this company owns approximately 10610^6 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
  1. 103810^{38}, because 1032⋅106=1032+6=103810^{32} \boldcdot 10^{6} = 10^{32+6} = 10^{38}.
  2. 10810^8 or $100,000,000.
Lesson 3
Powers of Powers of 10
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Making a Million (1 problem)

Here are some equivalent ways of writing 10410^4:

  • 10,000

  • 10⋅10310 \boldcdot 10^3

  • (102)2(10^2)^2

Write as many expressions as you can that have the same value as 10610^6.

Show Solution

Answers vary. Sample responses:

  • 1,000,000
  • 1,000⋅1,0001,000\boldcdot 1,000
  • 102⋅10410^2 \boldcdot 10^4
  • (103)2(10^3)^2
  • 1⋅1061\boldcdot 10^6
  • 10⋅10⋅10⋅10⋅10⋅1010\boldcdot10\boldcdot10\boldcdot10\boldcdot10\boldcdot10
     
Lesson 4
Dividing Powers of 10
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Why Subtract? (1 problem)

Why is 1015104\dfrac{10^{15}}{10^4} equal to 101110^{11}? Explain or show your thinking.

Show Solution

Sample response: 1015104=1011\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, 1015104=104⋅1011104=1011.\displaystyle \frac{10^{15}}{10^4} = \frac{10^4 \boldcdot 10^{11}}{10^4}=10^{11}.

Lesson 5
Negative Exponents with Powers of 10
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Negative Exponent True or False (1 problem)

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

  1. 10-5=-10510^{\text -5} = \text -10^5
  2. (102)-3=(10-2)3(10^2)^{\text -3} = (10^{\text -2})^3
  3. 1031014=10-11\dfrac{10^3}{10^{14}} = 10^{\text-11}
Show Solution
  1. False. Sample reasoning: 10-5=1100,00010^{\text-5} = \frac{1}{100,000}, whereas -105=-100,000\text-10^5 = \text-100,000.
  2. True. Sample reasoning: Both (102)-3\left(10^2 \right)^{\text-3} and (10-2)3\left(10^{\text-2} \right)^{3} are equal to 10-610^{\text-6}.
  3. True. Sample reasoning: 1031014=103−14=10-11\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 10-310^{\text-3}.
Show Solution
B, C, G
Problem 2

Write each expression using a single exponent.

  1. 105⋅(104)3107\dfrac{10^5\boldcdot(10^4)^3}{10^7}
  2. 100⋅101⋅1011(102)3\dfrac{10^0\boldcdot10^1\boldcdot10^{11}}{(10^2)^3}
Show Solution
  1. 101010^{10}
  2. 10610^6
Lesson 6
What about Other Bases?
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Spot the Mistake (1 problem)
  1. Diego was trying to write 23⋅222^3 \boldcdot 2^2 with a single exponent and wrote 23⋅22=23⋅2=262^3 \boldcdot 2^2 = 2^{3 \boldcdot 2} = 2^6. Do you agree with Diego? Explain your reasoning.
  2. Andre was trying to write 747-3\dfrac{7^4}{7^{\text -3}} with a single exponent and wrote 747-3=74−3=71\dfrac{7^4}{7^{\text -3}} = 7^{4-3} = 7^1. Do you agree with Andre? Explain your reasoning.
Show Solution
  1. 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: 23⋅22=(2⋅2⋅2)(2⋅2)=23+2=252^3 \boldcdot 2^2 = (2 \boldcdot 2 \boldcdot 2)(2 \boldcdot 2) = 2^{3 +2} = 2^5.
  2. I do not agree with Andre. Sample reasoning: Andre did 74−37^{4-3} when he should have done 74−(-3)7^{4-(\text- 3)} to get 777^7.
Lesson 7
Practice with Rational Bases
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Working with Exponents (1 problem)

  1. Rewrite each expression using a single, positive exponent:

    1. 9399\dfrac{9^3}{9^9}
    2. 14-3⋅141214^{\text-3} \boldcdot 14^{12}
  2. Diego wrote 64⋅83=4876^4 \boldcdot 8^3 = 48^7. Explain what Diego’s mistake was and how you know the equation is not true.
Show Solution
    1. 196\frac{1}{9^6}
    2. 14914^9
  2. 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
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Equivalent or Not? (1 problem)

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

  1. 127⋅4712^7 \boldcdot 4^7
  2. 12⋅4712 \boldcdot 4^7
  3. (12⋅7)⋅(4⋅7)(12\boldcdot 7)\boldcdot (4\boldcdot 7)
  4. 48748^7
Show Solution
  1. 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⋅4)(12\boldcdot 4).
  2. Not equivalent. Sample reasoning: This expression has only 1 factor that is 12 instead of 7 factors that are 12.
  3. 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.
  4. Equivalent. Sample reasoning: Since 12⋅4=4812\boldcdot 4=48 these expressions are equivalent.
Section B Check
Section B Checkpoint
Problem 1
Select all the expressions that are equivalent to 252^{5}.
Show Solution
A, E, F, H
Problem 2
Write 45⋅754^5\boldcdot7^5 using a single exponent.

Show Solution
28528^5
Problem 3
Explain why 34⋅833^4\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⋅83\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
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Better with Powers of 10 (1 problem)
  1. Write 0.000000123 as a multiple of a power of 10.
  2. Write 123,000,000 as a multiple of a power of 10.
Show Solution
  1. Sample response: (1.23)⋅10-7(1.23) \boldcdot 10^\text{-7} (or equivalent)
  2. Sample response: (1.23)⋅108(1.23) \boldcdot 10^8 (or equivalent)
Lesson 10
Representing Large Numbers on the Number Line
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Describe the Point (1 problem)

The speed of light through ice can be written as a multiple of a power of 10, such as (2.3)⋅108(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 AA and BB.

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.

  1. Describe point AA as:

    1. A multiple of a power of 10
    2. A value

  2. Describe point BB as:

    1. A multiple of a power of 10
    2. A value

  3. Plot a point CC that is greater than AA and less than BB. Describe point CC as:

    1. A multiple of a power of 10
    2. A value
Show Solution
    1. 2⋅1082 \boldcdot 10^8.
    2. 200,000,000.
    1. (5.3)⋅108(5.3) \boldcdot 10^8.
    2. 530,000,000.
  3. Sample response: any value between 2⋅1082 \boldcdot 10^8 and (5.3)⋅108(5.3) \boldcdot 10^8.
Lesson 11
Representing Small Numbers on the Number Line
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Describing Very Small Numbers (1 problem)
  1. Write 0.00034 as a multiple of a power of 10
  2. Write (5.64)⋅10-7(5.64) \boldcdot 10^{\text -7} as a decimal.
Show Solution
  1. (3.4)⋅10-4(3.4) \boldcdot 10^{\text-4}, 34⋅10-534 \boldcdot 10^{\text-5}, or equivalent
  2. 0.000000564
Lesson 12
Applications of Arithmetic with Powers of 10
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That's a Lot of Cells (1 problem)

There are about 260 million adults in the United States and the average adult has 10 pints of blood. If there are 2.4×10122.4\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 6.24×10216.24\times10^{21} red blood cells since (2.6×108)⋅(10)⋅(2.4×1012)=6.24×1021(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⋅1062\boldcdot10^6 square kilometers.
    1. The diameter of one atom of hydrogen is 5⋅10-115\boldcdot10^{\text-11} meters.
    2. The diameter of one atom of magnesium is 3⋅10-103\boldcdot10^{\text-10} meters.
Show Solution
  1. The country of Canada covers an area about 5 times the size of the country of Mexico, because 10,000,000=10⋅10610,000,000=10\boldcdot10^6, which is 5 times larger than 2⋅1062\boldcdot10^6.
  2. The diameter of one atom of magnesium is 6 times larger than the diameter of one atom of hydrogen, because 3⋅10-10=30⋅10-113\boldcdot10^{\text-10}=30\boldcdot10^{\text-11}, which is 6 times larger than 5⋅10-115\boldcdot10^{\text-11}
Problem 2
Fingernails grow at a rate of about 10-910^{\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
10710^7 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 10-910^{\text-9} meter per second, it will take 10910^9 seconds to grow 1 meter. Since there are 100 or 10210^2 centimeters in 1 meter, 109102=107\frac{10^9}{10^2}=10^7.
Lesson 13
Definition of Scientific Notation
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Scientific Notation Check (1 problem)

Determine which of the following numbers are written in scientific notation. If a number is not, write it in scientific notation.

  1. 5.23 ×1085.23 \times 10^8
  2. 48,200
  3. 0.00099
  4. 36×10536 \times 10^5
  5. 8.7 ×10-128.7 \times 10^{\text-12}
  6. 0.78 ×10-30.78 \times 10^{\text-3}
Show Solution
  1. Already in scientific notation
  2. 4.82×1044.82 \times 10^4
  3. 9.9×10-49.9 \times 10^\text{-4}
  4. 3.6×1063.6 \times 10^6
  5. Already in scientific notation
  6. 7.8×10-47.8\times 10^\text{-4}
Lesson 14
Estimating with Scientific Notation
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Estimating with Scientific Notation (1 problem)
  1. Estimate how many times larger 6.1 ×1076.1 \times 10^7 is than 2.1 ×10-42.1 \times 10^{\text -4}. Explain or show your reasoning.
  2. Estimate how many times larger 1.9 ×10-81.9 \times 10^{\text -8} is than 4.2 ×10-134.2 \times 10^{\text -13}. Explain or show your reasoning.
Show Solution
  1. 6.1×1076.1 \times 10^7 is about 300 billion times larger than 2.1×10-42.1 \times 10^{\text-4}. Sample reasoning: 6.1×1072.1×10-4≈6 ×1072 ×10-4=3×107−(-4)=3×1011.\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}.
  2. 1.9 ×10-81.9 \times 10^{\text-8} is about 50,000 times larger than 4.2 ×10-134.2 \times 10^{\text-13}. Sample reasoning: 1.9 ×10-84.2 ×10-13≈2 ×10-84 ×10-13=0.5×105=5×104.\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.
Lesson 15
Adding and Subtracting with Scientific Notation
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Adding with Scientific Notation (1 problem)

Elena wants to add (2.3 ×105)+(3.6 ×106)(2.3 \times 10^5) + (3.6 \times 10^6) and writes (2.3 ×105)+(3.6 ×106)=5.9 ×106(2.3 \times 10^5) + (3.6 \times 10^6) = 5.9 \times 10^6.

Explain to Elena what her mistake was and what the correct solution is.

Show Solution

Sample response: Elena added 2.3 and 3.6 without realizing that 3.6×1063.6 \times 10^6 is over 10 times as large as 2.3×1052.3 \times 10^5. Instead, she should have first rewritten one of the expressions so that both were multiplied by the same power of 10. For example, rewriting 3.6×1063.6 \times 10^6 as a multiple of 10510^5 gives 36×105+2.3×105=(36+2.3)×105=38.3×105=3.83×10636 \times 10^5 + 2.3 \times 10^5 = (36+2.3)\times 10^5 = 38.3 \times 10^5 = 3.83 \times 10^6.

Section D Check
Section D Checkpoint
Problem 1
Select all expressions written using scientific notation:
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 7.5×1097.5 \times 10^9 6.2×1016.2 \times 10^1
cows 1.3×1091.3 \times 10^9 4×1024 \times 10^2
sheep 1.75×1091.75 \times 10^9 6×1016 \times 10^1
chickens 2.4×10102.4 \times 10^{10} 2×1002 \times 10^0
ants 5×10165 \times 10^{16} 3×10-63 \times 10^{\text -6}
blue whales 4.7×1034.7 \times 10^3 1.9×1051.9 \times 10^5
antarctic krill 7.8×10147.8 \times 10^{14} 4.86×10-44.86 \times 10^{\text -4}
zooplankton 1×10201 \times 10^{20} 5×10-85 \times 10^{\text -8}
bacteria 5×10305 \times 10^{30} 1×10-121 \times 10^{\text -12}
 
  1. What is the total mass of 1 human, 1 cow, and 2 chickens? Explain or show your reasoning.

  2. What is the total mass of all the zooplankton on the planet? Explain or show your reasoning.
Show Solution
  1. 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=46662+400+2+2=466 kg.
  2. 5×10125\times10^{12} kg (or equivalent). Sample reasoning: (1×1020)×(5×10-8)=(1×5)×(1020×10-8)=5×1012(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?
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No cool-down
Unit 7 Assessment
End-of-Unit Assessment