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Lesson 14Multiplying, Dividing, and Estimating with Scientific Notation

Let’s multiply and divide with scientific notation to answer questions about animals, careers, and planets.

Learning Targets:

  • I can multiply and divide numbers given in scientific notation.
  • I can use scientific notation and estimation to compare very large or very small numbers.

14.1 True or False: Equations

Is each equation true or false? Explain your reasoning.

4\times 10^5 \times 4 \times 10^4= 4\times 10^{20}

\dfrac{7 \times 10^6}{2 \times 10^4} = (7\div2) \times 10^{(6-4)}

8.4 \times 10^3 \times 2 = (8.4 \times 2) \times 10^{(3 \times 2)}

14.2 Biomass

Use the table to answer questions about different creatures on the planet. Be prepared to explain your reasoning.

creature 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}
  1. Which creature is least numerous? Estimate how many times more ants there are.
  2. Which creature is the least massive? Estimate how many times more massive a human is.
  3. Which is more massive, the total mass of all the humans or the total mass of all the ants? About how many times more massive is it?
  4. Which is more massive, the total mass of all the krill or the total mass of all the blue whales? About how many times more massive is it?

14.3 Distances in the Solar System

Use the table to answer questions about the Sun and the planets of the solar system (sorry, Pluto).

object distance to Earth (km) diameter (km) mass (kg)
Sun 1.46 \times 10^8 1.392 \times 10^6 1.989 \times 10^{30}
Mercury 7.73 \times 10^7 4.878 \times 10^3 3.3 \times 10^{23}
Venus 4 \times 10^7 1.21 \times 10^4 4.87 \times 10^{24}
Earth N/A 1.28 \times 10^4 5.98 \times 10^{24}
Mars 5.46 \times 10^7 6.785 \times 10^3 6.4 \times 10^{23}
Jupiter 5.88 \times 10^8 1.428 \times 10^5 1.898 \times 10^{27}
Saturn 1.2 \times 10^9 1.199 \times 10^5 5.685 \times 10^{26}
Uranus 2.57 \times 10^9 5.149 \times 10^4 8.68 \times 10^{25}
Neptune 4.3 \times 10^9 4.949 \times 10^4 1.024 \times 10^{26}

Answer the following questions about celestial objects in the solar system. Express each answer in scientific notation and as a decimal number.

  1. Estimate how many Earths side by side would have the same width as the Sun.

  2. Estimate how many Earths it would take to equal the mass of the Sun.

  3. Estimate how many times as far away from Earth the planet Neptune is compared to Venus.

  4. Estimate how many Mercuries it would take to equal the mass of Neptune.

Are you ready for more?

Choose two celestial objects and create a scale image of them in the applet below.

object distance to Earth (km) diameter (km) mass (kg)
Sun (1.46) \boldcdot 10^8 (1.392) \boldcdot 10^6 (1.989) \boldcdot 10^{30}
Mercury (7.73) \boldcdot 10^7 (4.878) \boldcdot 10^3 (3.3) \boldcdot 10^{23}
Venus 4 \boldcdot 10^7 (1.21) \boldcdot 10^4 (4.87) \boldcdot 10^{24}
Earth N/A (1.28) \boldcdot 10^4 (5.98) \boldcdot 10^{24}
Mars (5.46) \boldcdot 10^7 (6.785) \boldcdot 10^3 (6.4) \boldcdot 10^{23}
Jupiter (5.88) \boldcdot 10^8 (1.428) \boldcdot 10^5 (1.898) \boldcdot 10^{27}
Saturn (1.2) \boldcdot 10^9 (1.199) \boldcdot 10^5 (5.685) \boldcdot 10^{26}
Uranus (2.57) \boldcdot 10^9 (5.149) \boldcdot 10^4 (8.68) \boldcdot 10^{25}
Neptune (4.3) \boldcdot 10^9 (4.949) \boldcdot 10^4 (1.024) \boldcdot 10^{26}

Plot a point

 for the center of each circle. Select the Circle with Center and Radius tool 
 and click on a point. When the dialog box opens, enter the radius.
open applet in presentation mode

14.4 Professions in the United States

Use the table to answer questions about professions in the United States as of 2012.

profession number typical annual salary (U.S. dollars)
architect 1.074 \times 10^5 7.3 \times 10^4
artist 5.14 \times 10^4 4.4 \times 10^4
programmer 1.36 \times 10^6 8.85 \times 10^4
doctor 6.9 \times 10^5 1.87 \times 10^5
engineer 6.17 \times 10^5 8.6 \times 10^4
firefighter 3.07 \times 10^5 4.5 \times 10^4
military—enlisted 1.16 \times 10^6 4.38 \times 10^4
military—officer 2.5 \times 10^5 1 \times 10^5
nurse 3.45 \times 10^6 6.03 \times 10^4
police officer 7.8 \times 10^5 5.7 \times 10^4
college professor 1.27 \times 10^6 6.9 \times 10^4
retail sales 4.67 \times 10^6 2.14 \times 10^4
truck driver 1.7 \times 10^6 3.82\times 10^4

Answer the following questions about professions in the United States. Express each answer in scientific notation.

  1. Estimate how many times more nurses there are than doctors.
  2. Estimate how much money all doctors make put together.
  3. Estimate how much money all police officers make put together.
  4. Who makes more money, all enlisted military put together or all military officers put together? Estimate how many times more.

Lesson 14 Summary

Multiplying numbers in scientific notation extends what we do when we multiply regular decimal numbers. For example, one way to find  (80)(60) is to view 80 as 8 tens and to view 60 as 6 tens. The product (80)(60) is 48 hundreds or 4,800. Using scientific notation, we can write this calculation as (8 \times 10^1) (6 \times 10^1) = 48 \times 10^2.  To express the product in scientific notation, we would rewrite it as  4.8 \times 10^3 .

Calculating using scientific notation is especially useful when dealing with very large or very small numbers. For example, there are about 39 million or 3.9 \times 10^7 residents in California. Each Californian uses about 180 gallons of water a day. To find how many gallons of water Californians use in a day, we can find the product (180) (3.9 \times 10^7) = 702 \times 10^7 , which is equal to 7.02 \times 10^9 . That’s about 7 billion gallons of water each day!

Comparing very large or very small numbers by estimation also becomes easier with scientific notation. For example, how many ants are there for every human? There are 5 \times 10^{16} ants and 7 \times 10^9 humans. To find the number of ants per human, look at \frac{5 \times 10^{16}}{7 \times 10^9} . Rewriting the numerator to have the number 50 instead of 5, we get  \frac{50 \times 10^{15}}{7 \times 10^9} . This gives us \frac{50}{7} \times 10^6 . Since \frac{50}{7} is roughly equal to 7, there are about 7 \times 10^6 or 7 million ants per person!

Lesson 14 Practice Problems

  1. Evaluate each expression. Use scientific notation to express your answer.

    1. (1.5 \times 10^2) (5 \times 10^{10})
    2. \frac{4.8 \times 10^{\text-8}}{3 \times 10^{\text-3}}
    3. (5 \times 10^8) (4 \times 10^3)
    4. (7.2 \times 10^3) \div (1.2 \times 10^5)

  2. How many bucketloads would it take to bucket out the world’s oceans? Write your answer in scientific notation.

    Some useful information:

    • The world’s oceans hold roughly 1.4 \times 10^{9} cubic kilometers of water.
    • A typical bucket holds roughly 20,000 cubic centimeters of water.
    • There are 10^{15} cubic centimeters in a cubic kilometer.

  3. The graph represents the closing price per share of stock for a company each day for 28 days.

    The graph represents the closing price per share of stock for a company each day for 28 days.
    1. What variable is represented on the horizontal axis?
    2. In the first week, was the stock price generally increasing or decreasing?
    3. During which period did the closing price of the stock decrease for at least 3 days in a row?

  4. Write an equation for the line that passes through (\text- 8.5, 11) and (5, \text- 2.5) .

  5. Explain why triangle ABC is similar to triangle CFE . 

    Traingles ABC and CEF are formed off of a linear function on a graph