# Operations with Numbers in Scientific Notation

## Add, subtract, multiply and divide values in scientific notation.

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Practice Operations with Numbers in Scientific Notation

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Solve Percent Problems Involving Scientific Notation

On certain dates, Mars is about \begin{align*}4.9 \times 10^7\end{align*} miles from Earth. If a spacecraft is headed toward Mars and has traveled 30% of the distance, how many miles has it gone?

In this concept, you will learn to solve percent problems involving scientific notation.

### Scientific Notation

Scientific notation is a useful mathematical tool that allows you to work with very large or very small numbers.  Scientific notation is when a number is written as a rational number and a power of 10. This means that you are using exponents to represent the power of 10. Scientific notation has the general form: \begin{align*}a \times 10^b\end{align*}

where ‘\begin{align*}a\end{align*}’ is any number between 1 and 10, and ‘\begin{align*}b\end{align*}’ is the exponent of 10.

When you conduct operations involving percent with numbers in scientific notation, you can use any operation with the ‘\begin{align*}a\end{align*}’ value as you have seen in these sections. Then you will be sure to write your answer in scientific notation. You may need to adjust the ‘\begin{align*}a\end{align*}’ and ‘\begin{align*}b\end{align*}’ values.

Let’s look at an example.

Find 25% of \begin{align*}3 \times 10^{12}\end{align*}.

First, multiply the 25% by 3.

\begin{align*}\begin{array}{rcl} 0.25 \times (3 \times 10^{12}) &=& (0.25 \times 3) \times 10^{12} \\ &=& 0.75 \times 10^{12} \end{array} \end{align*}

Next, since the ‘\begin{align*}a\end{align*}’ value is not between 1 and 10, you must move the decimal place over one place. In other words, 0.75 must change to 7.5. Since you are moving the decimal one place to the right, your exponent \begin{align*}b\end{align*} will go down by 1.

\begin{align*}0.75 \times 10^{12} = 7.5 \times 10^{11}\end{align*}

The answer is \begin{align*}7.5 \times 10^{11}\end{align*}.

Very large and very small numbers are not always written in scientific notation. Some numbers remain expanded or in standard notation. You can still work with numbers that are very large but again must be most careful of decimal places.

Let’s look at an example.

In the year 2000, the United States had a population of about 280,000,000 people. By 2010, the population was about 308,000,000. What will the percent increase have been in those 10 years?

First, find the difference in the populations.

\begin{align*}308,000,000-280,000,000=28,000,000\end{align*}

Next, use the percent increase formula to find the answer.

\begin{align*}\begin{array}{rcl} \text{percent of increase} &=& \frac{\text{amount of increase}}{\text{original amount}}\times 100 \\ \text{percent of increase} &=& \frac{28,000,000}{280,000,000} \times 100 \\ \text{percent of increase} &=& 0.10 \times 100 \\ \text{percent of increase} &=& 10 \% \end{array}\end{align*}

The population increased 10% in 10 years.

### Examples

#### Example 1

Earlier, you were given a problem about the trip to Mars.

You want to find the number of miles a spacecraft has gone if it is 30% of the way to Mars, which at its nearest is about \begin{align*}4.9 \times 10^7\end{align*} miles from Earth.

First, change 30% into a decimal.

\begin{align*}\begin{array}{rcl} 30 \% &=& \frac{30}{100} \\ &=& 0.30 \end{array}\end{align*}

Next, multiply 0.30 by \begin{align*}4.9 \times 10^7\end{align*}

\begin{align*}\begin{array}{rcl} 0.30 \times (4.9 \times 10^7) &=& (0.30 \times 4.9) \times 10^7 \\ &=& 1.47 \times 10^7 \end{array}\end{align*}

The answer is \begin{align*}1.47 \times 10^7\end{align*}.

The spacecraft has traveled \begin{align*}1.47 \times 10^7\end{align*} miles towards Mars if it is 30% of the way.

#### Example 2

Find 32% of 0.00000054.

First, change 32% into a decimal. Remember that percent means that the denominator is 100.

\begin{align*}\begin{array}{rcl} 32 \% &=& \frac{32}{100} \\ &=& 0.32 \end{array} \end{align*}

Next, multiply 0.32 by 0.00000054

\begin{align*}0.32 \times 0.00000054 =0.0000001728\end{align*}

Then, change to scientific notation. Notice that the number is smaller than 1 so the exponent \begin{align*}(b)\end{align*} will be negative. Move the decimal place until it is after the 1. This means you are moving the decimal 7 places to the right; your exponent \begin{align*}(b)\end{align*} is -7.

\begin{align*}0.0000001728 = 1.728 \times 10^{-7}\end{align*}

The answer is \begin{align*}1.728 \times 10^{-7}\end{align*}.

#### Example 3

Find 30% of .000567

First, change 30% into a decimal. Remember that percent means that the denominator is 100.

\begin{align*}\begin{array}{rcl} 30 \% &=& \frac{30}{100} \\ &=& 0.30 \end{array} \end{align*}

Next, multiply 0.30 by 0.000567

\begin{align*}0.30 \times 0.000567 =0.0001701\end{align*}

Then, change to scientific notation. Notice that the number is smaller than 1 so the exponent \begin{align*}(b)\end{align*} will be negative. Move the decimal place until it is after the 1. This means you are moving the decimal 4 places to the right; your exponent \begin{align*}(b)\end{align*} is -4.

\begin{align*}0.0001701 = 1.701 \times 10^{-4}\end{align*}

The answer is \begin{align*}1.701 \times 10^{-4}\end{align*}.

#### Example 4

Find 10% of 123,000,000

First, change 10% into a decimal.

\begin{align*} \begin{array}{rcl} 10 \% &=& \frac{10}{100} \\ &=& 0.10 \end{array}\end{align*}

Next, multiply 0.10 by 123,000,000

\begin{align*}0.10 \times 123,000,000 =12,300,000\end{align*}

Then, change to scientific notation. Notice that the number is larger than 1 so the exponent \begin{align*}(b)\end{align*} will be positive. Move the decimal place until it is after the 1. This means you are moving the decimal 7 places to the left; your exponent \begin{align*}(b)\end{align*} is 7.

\begin{align*}12,300,000 = 1.23 \times 10^7\end{align*}

The answer is \begin{align*}1.23 \times 10^7\end{align*}.

#### Example 5

Find 25% of 0.0000987.

First, change 25% into a decimal.

\begin{align*}\begin{array}{rcl} 25 \% &=& \frac{25}{100} \\ &=& 0.25 \end{array}\end{align*}

Next, multiply 0.25 by 0.0000987

\begin{align*}0.25 \times 0.0000987 =0.000024675\end{align*}

Then, change to scientific notation. Notice that the number is smaller than 1 so the exponent \begin{align*}(b)\end{align*} will be negative. Move the decimal place until it is after the 1. This means you are moving the decimal 5 places to the right; your exponent \begin{align*}(b)\end{align*} is -5.

\begin{align*}0.000024675 = 2.4675 \times 10^{-5}\end{align*}

The answer is \begin{align*}2.47 \times 10^{-5}\end{align*}.

### Review

Round the ‘\begin{align*}a\end{align*}’ values to the nearest hundredth and place your answers in scientific notation.

1. Find 62% of \begin{align*}3.5 \times 10^9\end{align*}.

2. Find 5% of \begin{align*}9.1 \times 10^{13}\end{align*}.

3. Find 180% of \begin{align*}6.3 \times 10^{-17}\end{align*}.

4. Find 12% of \begin{align*}.18 \times 10^3\end{align*}.

5. Find 22% of \begin{align*}56.4 \times 10^{-2}\end{align*}.

6. Find 14% of \begin{align*}56.4 \times 10^{-2}\end{align*}.

7. Find 30% of \begin{align*}.999 \times 10^{12}\end{align*}.

8. What percent of 8,570,000 is 152?

9. Find 230% of .00000488.

10. The number .00036 is 45% of what number?

11. Find 23% of 98.78.

12. Find 150% of .0000866.

13. Find 210% of .002368.

14. Find 30% of .000009.

15. A light year is about 5,880,000,000,000 miles. In one month, it travels about 8.2% of that distance. About how far does it travel in one month?

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### Vocabulary Language: English

TermDefinition
Percent of Change The percent of change is the percent that a value has increased or decreased by.
Standard Notation As opposed to scientific notation, standard notation means writing numbers in the usual way with all of the zeros accounted for in the value.
Statistics Statistics is a branch of mathematics that involves collecting, analyzing and displaying data.