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2.18: Operations with Numbers in Scientific Notation

Difficulty Level: At Grade Created by: CK-12
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Remember Kara's study of the solar system?

Well, after gathering information, Kara decided to add some distances together. She decided to add the distances from Earth to Saturn with the distance from Earth to Jupiter.

Here is what she wrote.

5.95×108+8.87×108

Do you know the sum?

This Concept will teach you how to add, subtract, multiply and divide values written in scientific notation.

Guidance

Scientific notation makes reading and writing very large and very small numbers easier; it makes computation with such numbers easier as well.

Let’s start with addition and subtraction. Before performing addition or subtraction on scientific notation, the exponents must be the same. Matching the exponents involves a simple case of moving the decimal point—a process you’ve completed many times in making the divisor a whole number before dividing decimals. Let’s see how it’s done in the following addition problem. Note how we use parentheses to group the scientific notation on either side of the addition sign.

(5.7×104)+(4.87×105)

We want to make both of these exponents the same. To make both exponents 5’s, we move the decimal point in 5.7 one place to the left by multiplying by 10.

(.57×105)+(4.87×105)

Now we can add the decimal parts of the problem. The power of 10 stays the same.

(.57×105)+(4.87×105)(.57+4.87)×1055.44×105

Our answer is 5.44×105.

Subtraction works the same way as addition: Before performing the subtraction operation, the exponents must be the same.

Multiplication and division in scientific notation is a little different.

Do you remember simplifying exponents?

(x3)x4

To multiply the exponents, we add the powers. (x3)x4=x3+4=x7. Multiplying scientific notation is similar: You multiply the decimals and add the exponents.

(3.4×102)×(6.2×106)(3.4×6.2)×(102+6)21.08×104

Division of scientific notation is identical to multiplication—except you divide the decimals and subtract the exponents. Let’s try it out.

(8.4×105)÷(1.4×102)(8.4÷1.4)×(105(2))Remember subtracting a negative is the same as adding it.6×107

Now it's time for you to try a few on your own.

Example A

Add (3.4×103+5.6×104)

Solution: 39.6×104

Example B

Multiply (1.2×104)(3.4×104)

Solution:4.08×108

Example C

Subtract (5.6×1043.2×104)

Solution: 2.4×104

Here is the original problem once again.

Well, after gathering information, Kara decided to add some distances together. She decided to add the distances from Earth to Saturn with the distance from Earth to Jupiter.

Here is what she wrote.

5.95×108+8.87×108

Do you know the sum?

First, notice that the exponent is the same in both values. Therefore, we can simply add the decimals.

5.95+8.87=14.82

Now we add the rest of the scientific notation.

14.82×108

This is our answer.

Vocabulary

Here are the vocabulary words in this Concept.

Standard Form
the writing of a number with zeros not written using exponents and powers of 10.
Exponential Form
A number written with an exponent
Scientific Notation
Numbers that are written as decimal products with base ten powers

Guided Practice

Here is one for you to try on your own.

At its closest, the planet Neptune is 4,300,000,000 kilometers away from Earth. A group of astronauts from Earth want to make it to Neptune is 20,000 days. If they travel the same amount of kilometers each day, how many kilometers will they travel each day? Convert both numbers to scientific notation before solving.

Answer

Let’s begin by converting both numbers to scientific notation.

The distance between Earth and Neptune, in scientific notation, is 4.3×109. The number of days the astronauts want to travel in scientific notation is 2.0×104.

We want to divide the distance evenly among the days, so we know we need to divide. Remember: To divide numbers in scientific notation, you divide the decimals and subtract the exponents.

4.3×109÷2.0×104(4.3÷2.0)×10942.15×105

Remember to put the units of measurement in your answer!

Our answer is 2.15×105 km or 215,000 kilometers.

Video Review

Here are videos for review.

- This is a James Sousa video about writing decimal numbers given scientific notation.

- This is a James Sousa video about multiplying values that are in scientific notation.

- This is a James Sousa video about dividing numbers that are in scientific notation.

Practice

Directions: Add, subtract, multiply or divide

1. 3.4×103+5.4×103

2. 5.4×1041.3×104

3. 6.7×105+5.4×105

4. 13.4×1035.4×103

5. 6.4×103×5.1×103

6. 12.4×103÷2.2×103

7. 5.4×104+4.4×105

8. 12.2×10210.1×103

9. 5.6×103+4.5×103

10. 3.3×104×1.2×102

11. 24.6×105÷6.1×103

12. 266×104+8.6×106

13. 7.14×1045.5×103

14. (2.56×103)×(3.8×106)

15. (4.97×108)÷(7.9×105)

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Date Created:
Nov 30, 2012
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Feb 26, 2016
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