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# 8.5: Scientific Notation

Difficulty Level: Basic Created by: CK-12
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Did you know that the average distance of the Earth from the Sun is about 92,000,000 miles? This is a big number! Do you think that there is any way to write it more compactly? In this Concept, you'll learn all about using scientific notation so that you can express very large or very small numbers as the product of a decimal and 10 to a certain power. This way, you'll be able to express numbers such as 92,000,000 miles much more succinctly.

### Guidance

Sometimes in mathematics numbers are huge. They are so huge that we use what is called scientific notation. It is easier to work with such numbers when we shorten their decimal places and multiply them by 10 to a specific power. In this Concept, you will learn how to express numbers using scientific notation.

Definition: A number is expressed in scientific notation when it is in the form

N×10n\begin{align*}N \times 10^n\end{align*}

where 1N<10\begin{align*}1\le N <10\end{align*} and n\begin{align*} n \end{align*} is an integer.

For example, 2.35×1037\begin{align*} 2.35 \times 10^{37}\end{align*} is a number expressed in scientific notation. Notice there is only one number in front of the decimal place.

Since the scientific notation uses powers of ten, we want to be comfortable expressing different powers of ten.

Powers of 10:

100,00010,0001,00010010=105=104=103=102=101\begin{align*}100,000 &= 10^5\\ 10,000 &= 10^4\\ 1,000 &= 10^3\\ 100 &= 10^2\\ 10 &= 10^1\end{align*}

Using Scientific Notation for Large Numbers

If we divide 643,297 by 100,000 we get 6.43297. If we multiply 6.43297 by 100,000, we get back to our original number, 643,297. But we have just seen that 100,000 is the same as 105\begin{align*}10^5\end{align*}, so if we multiply 6.43297 by 105\begin{align*}10^5\end{align*}, we should also get our original number, 643,297, as well. In other words 6.43297×105=643,297\begin{align*}6.43297 \times 10^5=643,297\end{align*}. Because there are five zeros, the decimal moves over five places.

#### Example A

Look at the following examples:

2.08×1042.08×1032.08×1022.08×1012.08×100=20,800=2,080=208=20.8=2.08\begin{align*}2.08 \times 10^4 &= 20,800\\ 2.08 \times 10^3 &= 2,080\\ 2.08 \times 10^2 &= 208\\ 2.08 \times 10^1 &= 20.8\\ 2.08 \times 10^0 &= 2.08\end{align*}

The power tells how many decimal places to move; positive powers mean the decimal moves to the right. A positive 4 means the decimal moves four positions to the right.

#### Example B

Write in scientific notation.

653,937,000

Solution:

653,937,000=6.53937000×100,000,000=6.53937×108\begin{align*}653,937,000=6.53937000 \times 100,000,000=6.53937 \times 10^8\end{align*}

Oftentimes, we do not keep more than a few decimal places when using scientific notation, and we round the number to the nearest whole number, tenth, or hundredth depending on what the directions say. Rounding Example A could look like 6.5×108\begin{align*}6.5 \times 10^8\end{align*}.

Using Scientific Notation for Small Numbers

We’ve seen that scientific notation is useful when dealing with large numbers. It is also good to use when dealing with extremely small numbers.

#### Example C

Look at the following examples:

2.08×1012.08×1022.08×1032.08×104=0.208=0.0208=0.00208=0.000208\begin{align*}2.08 \times 10^{-1} &= 0.208\\ 2.08 \times 10^{-2} &= 0.0208\\ 2.08 \times 10^{-3} &= 0.00208\\ 2.08 \times 10^{-4} &= 0.000208\end{align*}

### Guided Practice

The time taken for a light beam to cross a football pitch is 0.0000004 seconds. Write in scientific notation.

Solution:

0.0000004=4×0.0000001=4×110,000,000=4×1107=4×107\begin{align*}0.0000004=4 \times 0.0000001=4 \times \frac{1}{10,000,000}=4 \times \frac{1}{10^7}=4 \times 10^{-7}\end{align*}

### Practice

Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Scientific Notation (14:26)

Write the numerical value of the following expressions.

1. 3.102×102\begin{align*}3.102 \times 10^2\end{align*}
2. 7.4×104\begin{align*}7.4 \times 10^4\end{align*}
3. 1.75×103\begin{align*}1.75 \times 10^{-3}\end{align*}
4. 2.9×105\begin{align*}2.9 \times 10^{-5}\end{align*}
5. 9.99×109\begin{align*}9.99 \times 10^{-9}\end{align*}

Write the following numbers in scientific notation.

1. 120,000
2. 1,765,244
3. 63
4. 9,654
5. 653,937,000
6. 1,000,000,006
7. 12
8. 0.00281
9. 0.000000027
10. 0.003
11. 0.000056
12. 0.00005007
13. 0.00000000000954

#### Quick Quiz

1. Simplify: (2x4y3)3  x3y22x0y2\begin{align*}\frac{(2x^{-4}y^3)^{-3} \ \cdot \ x^{-3} y^{-2}}{-2x^0y^2}\end{align*}.
2. The formula A=1,500(1.0025)t\begin{align*}A=1,500(1.0025)^t\end{align*} gives the total amount of money in a bank account with a balance of \$1,500.00, earning 0.25% interest, compounded annually. How much money would be in the account five years in the past?
3. True or false? (54)3=12564\begin{align*}\left(\frac{5}{4}\right)^{-3}= -\frac{125}{64}\end{align*}

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### Vocabulary Language: English Spanish

Scientific Notation

Scientific notation is a means of representing a number as a product of a number that is at least 1 but less than 10 and a power of 10.

order of magnitude

Formally, the order of magnitude is the exponent in scientific notation. Informally it refers to size. Two objects or numbers are of the same order of magnitude are relatively similar sizes.

Standard Form

As opposed to scientific notation, standard form means writing numbers in the usual way with all of the zeros accounted for in the value.

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