**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.

**Related Pages**

Illustrative Math

Grade 8

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

Illustrative Math Unit 8.7, Lesson 14 (printable worksheets)

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 × 10^{1})(6 × 10^{1}) = 48 × 10^{2}

To express the product in scientific notation, we would rewrite it as 4.8 × 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 × 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 × 10^{7}) = 702 × 10^{7}, which is equal to 7.02 × 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 × 10^{16} ants and 7 × 10^{9} humans. To find the number of ants per human, look at (5 × 10^{16})/(7 × 10^{9}). Rewriting the numerator to have the number 50 instead of 5, we get (50 × 10^{15})/(7 × 10^{9}). This gives us 50/7 × 10^{6}. Since 50/7 is roughly equal to 7, there are about 7 × 10^{6} or 7 million ants per person!

Is each equation true or false? Explain your reasoning.

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

- Which creature is least numerous? Estimate how many times more ants there are.
- Which creature is the least massive? Estimate how many times more massive a human is.
- 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?
- 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?

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

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

- Estimate how many Earths side by side would have the same width as the Sun.
- Estimate how many Earths it would take to equal the mass of the Sun.
- Estimate how many times as far away from Earth the planet Neptune is compared to Venus.
- Estimate how many Mercuries it would take to equal the mass of Neptune.

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

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

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

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

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

- Evaluate each expression. Use scientific notation to express your answer.
- 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 cubic kilometers of water.
- A typical bucket holds roughly 20,000 cubic centimeters of water.
- There are cubic centimeters in a cubic kilometer.

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

a. What variable is represented on the horizontal axis?

b. In the first week, was the stock price generally increasing or decreasing?

c. During which period did the closing price of the stock decrease for at least 3 days in a row? - Write an equation for the line that passes through (-8.5,11) and (5,-2.5).
- Explain why triangle ABC is similar to triangle CFE.

The Open Up Resources math curriculum is free to download from the Open Up Resources website and is also available from Illustrative Mathematics.

Try the free Mathway calculator and
problem solver below to practice various math topics. Try the given examples, or type in your own
problem and check your answer with the step-by-step explanations.

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