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# 2.18: Operations with Numbers in Scientific Notation

Difficulty Level: At Grade Created by: CK-12

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.

\begin{align*}5.95 \times 10^8 + 8.87 \times 10^8\end{align*}

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.

\begin{align*}(5.7 \times 10^4) + (4.87 \times 10^5)\end{align*}

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.

\begin{align*}(.57 \times 10^5) + (4.87 \times 10^5)\end{align*}

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

\begin{align*}&(.57 \times 10^5) + (4.87 \times 10^5)\\ &(.57 + 4.87) \times 10^5\\ & 5.44 \times 10^5\end{align*}

Our answer is \begin{align*}5.44 \times 10^5\end{align*}.

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?

\begin{align*}(x^3)x^4\end{align*}

To multiply the exponents, we add the powers. \begin{align*}(x^3)x^4 = x^{3 + 4} = x^7\end{align*}. Multiplying scientific notation is similar: You multiply the decimals and add the exponents.

\begin{align*}&(3.4 \times 10^{-2}) \times (6.2 \times 10^6)\\ & (3.4 \times 6.2) \times (10^{-2 + 6})\\ & 21.08 \times 10^4\end{align*}

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

\begin{align*}&(8.4 \times 10^5) \div (1.4 \times 10^{-2})\\ & (8.4 \div 1.4) \times (10^{5 - (-2)}) \rightarrow \text{Remember subtracting a negative is the same as adding it.}\\ & 6 \times 10^7\end{align*}

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

#### Example A

Add \begin{align*}(3.4 \times 10^3 + 5.6 \times 10^4)\end{align*}

Solution: \begin{align*}39.6 \times 10^4\end{align*}

#### Example B

Multiply \begin{align*}(1.2 \times 10^4)(3.4 \times 10^4)\end{align*}

Solution:\begin{align*}4.08 \times 10^8\end{align*}

#### Example C

Subtract \begin{align*}(5.6 \times 10^4 - 3.2 \times 10^4)\end{align*}

Solution: \begin{align*}2.4 \times 10^4\end{align*}

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.

\begin{align*}5.95 \times 10^8 + 8.87 \times 10^8\end{align*}

Do you know the sum?

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

\begin{align*}5.95 + 8.87 = 14.82\end{align*}

Now we add the rest of the scientific notation.

\begin{align*}14.82 \times 10^8\end{align*}

### 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.

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

The distance between Earth and Neptune, in scientific notation, is \begin{align*}4.3 \times 10^9\end{align*}. The number of days the astronauts want to travel in scientific notation is \begin{align*}2.0 \times 10^4\end{align*}.

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.

\begin{align*}&4.3 \times 10^9 \div 2.0 \times 10^4\\ & (4.3 \div 2.0) \times 10^{9 - 4}\\ & 2.15 \times 10^5\end{align*}

Our answer is \begin{align*}2.15 \times 10^5 \ km\end{align*} or 215,000 kilometers.

### Video Review

Here are videos for review.

### Practice

Directions: Add, subtract, multiply or divide

1. \begin{align*}3.4 \times 10^3 + 5.4 \times 10^3\end{align*}

2. \begin{align*}5.4 \times 10^4 - 1.3 \times 10^4\end{align*}

3. \begin{align*}6.7 \times 10^5 + 5.4 \times 10^5\end{align*}

4. \begin{align*}13.4 \times 10^3 - 5.4 \times 10^3\end{align*}

5. \begin{align*}6.4 \times 10^3 \times 5.1 \times 10^3\end{align*}

6. \begin{align*}12.4 \times 10^3 \div 2.2 \times 10^3\end{align*}

7. \begin{align*}5.4 \times 10^4 + 4.4 \times 10^5\end{align*}

8. \begin{align*}12.2 \times 10^2 - 10.1 \times 10^3\end{align*}

9. \begin{align*}5.6 \times 10^3 + 4.5 \times 10^3\end{align*}

10. \begin{align*}3.3 \times 10^4 \times 1.2 \times 10^2\end{align*}

11. \begin{align*}24.6 \times 10^5 \div 6.1 \times 10^3\end{align*}

12. \begin{align*}266 \times 10^{-4} + 8.6 \times 10^{-6}\end{align*}

13. \begin{align*}7.14 \times 10^4 - 5.5 \times 10^3\end{align*}

14. \begin{align*}(2.56 \times 10^{-3}) \times (3.8 \times 10^6)\end{align*}

15. \begin{align*}(4.97 \times 10^8) \div (7.9 \times 10^5)\end{align*}

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