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How to divide scientific notation

# How to divide scientific notation

Here you will learn about how to divide with scientific notation including what it is and how to solve problems.

Students will first learn how to divide with scientific notation as part of expressions and equations in 8 th grade.

## What is how to divide scientific notation?

Dividing with scientific notation is completing the division between two numbers that are written in scientific notation.

Scientific notation is writing numbers in this form:

a\times10^{n}

Where a is a number 1\leq{a}<10 and n is an integer (whole number).

Numbers written in scientific notation can make some calculations with very large numbers or small numbers neater and quicker to compute.

For example,

Now let’s solve \left(5.4\times{10^5}\right)\div\left(3\times{10^3}\right)

Re–write this expression as \cfrac{5.4\times{10^5}}{3\times{10^3}}, which equals \cfrac{5.4\times{10}\times{10}\times{10}\times{10}\times{10}}{3\times{10}\times{10}\times{10}}.

Notice how you can divide the corresponding parts to simplify.

This is the same as solving:

\begin{aligned}& \left(5.4 \times 10^5\right) \div\left(3 \times 10^3\right) \\\\ & =(5.4 \div 3) \times\left(10^5 \div 10^3\right) \\\\ & =1.8 \times 10^2 \end{aligned}

Since 1.8 is between 1 and 10, you don’t need to adjust the power of 10.

## Common Core State Standards

How does this relate to 8 th grade math?

• Grade 8 – Expressions and Equations (8.EE.A.3)
Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.

For example, estimate the population of the United States as 3\times{10^8} and the population of the world as 7\times{10^9}, and determine that the world population is more than 20 times larger.

• Grade 8 – Expressions and Equations (8.EE.A.4)
Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used.

Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (for example, use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.

## How to divide with scientific notation

In order to divide with scientific notation:

1. Divide the non-zero numbers.
2. Divide the powers of \bf{10} by subtracting the exponents.
3. Write the solution in scientific notation.

## How to divide scientific notation examples

### Example 1: non-zero numbers are whole numbers

1. Divide the non-zero numbers.

7\div{2}=3.5

2Divide the powers of \bf{10} by subtracting the exponents.

{10^7}\div{10^2}={10^{7-2}}={10^5}

3Write the solution in scientific notation.

3.5\times{10^5}

Since 3.5 is between 1 and 10, you don’t need to adjust the power of 10.

### Example 2: non-zero numbers are rational

Calculate 8.1 \times 10^9 \div 1.6 \times 10^4. Write your answer in scientific notation.

Divide the non-zero numbers.

Divide the powers of \bf{10} by subtracting the exponents.

Write the solution in scientific notation.

### Example 3: dividing by a decimal

Calculate 4 \times 10^3 \div 8 \times 10^{-5}. Write your answer in scientific notation.

Divide the non-zero numbers.

Divide the powers of \bf{10} by subtracting the exponents.

Write the solution in scientific notation.

### Example 4: a dividend with a negative exponent

Calculate 6.7 \times 10^{-2} \div 6.8 \times 10^{11}. Write your answer in scientific notation.

Divide the non-zero numbers.

Divide the powers of \bf{10} by subtracting the exponents.

Write the solution in scientific notation.

### Example 5: word problem

Al’s Jams social media account has 9.08 \times 10^2 subscribers. Manic Music’s account has 4.3 \times 10^5 subscribers. How many times more subscribers does Manic Music’s account have? Write your answer in scientific notation.

Divide the non-zero numbers.

Divide the powers of \bf{10} by subtracting the exponents.

Write the solution in scientific notation.

### Example 6: word problem

A factory produces 5.4 \times 10^3 candles per day. How many days will it take for them to complete an order of 1.8 \times 10^{4} candles?

Divide the non-zero numbers.

Divide the powers of \bf{10} by subtracting the exponents.

Write the solution in scientific notation.

### Teaching tips for how to divide scientific notation

• Before teaching students how to divide numbers with scientific notation, review exponential notation and exponent rules.

• When introducing the topic, start with worksheets that have practice problems with whole number coefficients before moving on to ones with decimal numbers.

• To prevent the skill of dividing decimals from holding students back, give them access to a decimal division tutorial when solving or let them use a scientific calculator.

### Easy mistakes to make

• Not converting solutions to scientific notation
After calculating with scientific notation, it is important to check that the first part is 1\leq{n}<10. If it is less than 1 or greater than 10, use powers of 10 to convert it.

• Incorrectly using the rules for powers of ten
When converting between powers of ten, remember that each place value is 10 times larger than the place to the right and 10 times smaller than the place to the left. This causes the decimal point to “move” as we divide or multiply by ten.

For example,
0.99 \times 10^{-13} has no digits left of the decimal. It needs to be converted.

=\left(9.9 \times 10^{-1}\right) \times 10^{-13} \quad \quad *Multiplying by 10^{-1} shifts the decimal places

\begin{aligned} & =9.9 \times\left(10^{-1} \times 10^{-13}\right) \\\\ & =9.9 \times\left(10^{-1+(-13)}\right) \\\\ & =9.9 \times 10^{-14} \end{aligned}

• Thinking that dividing in scientific notation always leads to smaller numbers
Just like dividing by a fraction, dividing by a decimal that is less than 1 will lead to a larger quotient.

For example,
\left(2 \times 10^3\right) \div\left(1 \times 10^{-1}\right)=2,000 \div 0.1=20,000

• Confusing the meaning of negative exponents
Every position to the right of the decimal is represented with powers of 10 that have negative exponents. Each has an equivalent fraction, where the numerator is 1 and the denominator is a power of ten (where the exponent corresponds to the number of places).

For example,
\begin{aligned}& 10^{-1}=\cfrac{1}{10} \\\\ &10^{-2}=\cfrac{1}{100} \\\\ &10^{-3}=\cfrac{1}{1,000} \\\\ &10^{-4}=\cfrac{1}{10,000} \end{aligned}

### Practice scientific notation questions

1. Solve \left(8 \times 10^7\right) \div\left(2 \times 10^2\right) . Write your answer in scientific notation.

40 \times 10^4

4\times{10^5}

0.5 \times 10^{-5}

2 \times 10^{3.5}

Rewrite this expression as \cfrac{8 \times 10^7}{2 \times 10^2}, which equals

\cfrac{8 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10}{2 \times 10 \times 10}.

Notice how you can divide the corresponding parts to simplify.

This is the same as solving:

\begin{aligned}& \left(8 \times 10^7\right) \div\left(2 \times 10^2\right) \\\\ & =(8 \div 2) \times\left(10^7 \div 10^2\right) \\\\ & =4 \times 10^5 \end{aligned}

2. Solve \left(9.65 \times 10^9\right) \div\left(3.1 \times 10^6\right) . Write your answer in scientific notation, rounded to the nearest hundredth.

6.55 \times 10^3

30.01 \times 10^{15}

3 \times 10^{1.5}

3.11 \times 10^3

Rewrite this expression as \cfrac{9.65 \times 10^9}{3.1 \times 10^6}, which equals

\cfrac{9.65 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10}{3.1 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10}.

Notice how you can divide the corresponding parts to simplify.

This is the same as solving:

\begin{aligned}& \left(9.65 \times 10^9\right) \div\left(3.1 \times 10^6\right) \\\\& =(9.65 \div 3.1) \times\left(10^9 \div 10^6\right) \\\\& =3.11 \times 10^3\end{aligned}

3. Solve \left(7.5 \times 10^4\right) \div\left(3.4 \times 10^{-8}\right) . Write your answer in scientific notation, rounded to the nearest hundredth.

2.21 \times 10^{12}

2.2 \times 10^{-4}

4.1 \times 10^4

0.45 \times 10^{-12}
\begin{aligned}& \left(7.5 \times 10^4\right) \div\left(3.4 \times 10^{-8}\right) \\\\ & =(7.5 \div 3.4) \times\left(10^4 \div 10^{-8}\right) \\\\ & =2.21 \times 10^{12} \end{aligned}

4. Solve \left(1.02 \times 10^5\right) \div\left(8 \times 10^{11}\right) . Write your answer in scientific notation, rounded to the nearest hundredth.

6.98 \times 10^{-6}

7.84 \times 10^7

1.275 \times 10^{-7}

4.6 \times 10^6
\begin{aligned}& \left(1.02 \times 10^5\right) \div\left(8 \times 10^{11}\right) \\\\ & =(1.02 \div 8) \times\left(10^5 \div 10^{11}\right) \\\\ & =0.1275 \times 10^{-6} \\\\ & =\left(1.275 \times 10^{-1}\right) \times 10^{-6} \\\\ & =1.275 \times 10^{-7} \end{aligned}

5. The Funky Jams playlist has been listened to 3.02 \times 10^6 times. The Bopping Beats playlist has been listened to 9.4 \times 10^3 times. How many times more listens does the Funky Jams playlist have? Write your answer in scientific notation, rounded to the nearest hundredth.

3.11 \times 10^3

3.11 \times 10^{-3}

6.38 \times 10^9

3.21 \times 10^2

To solve, divide 3.02 \times 10^6 by 9.4 \times 10^3.

\begin{aligned}& \left(3.02 \times 10^6\right) \div\left(9.4 \times 10^3\right) \\\\ & =(3.02 \div 9.4) \times\left(10^6 \div 10^3\right) \\\\ & =0.321 \times 10^3 \;\; (\text { rounded }) \\\\ & =\left(3.21 \times 10^{-1}\right) \times 10^3 \\\\ & =3.21 \times 10^2 \end{aligned}

The Funky Jams playlist has 3.21 \times 10^2 more listens than the Bopping Beats playlist.

6. A warehouse has 5.7 \times 10^8 pencils. Each school needs 8.3 \times 10^3 pencils for the new school year. How many schools can the warehouse provide pencils to? Write your answer in scientific notation, rounded to the nearest hundredth.

6.87 \times 10^4

2.6 \times 10^5

1.45 \times 10^{-5}

14 \times 10^{11}

To solve, divide 5.7 \times 10^8 by 8.3 \times 10^3.

\begin{aligned}& \left(5.7 \times 10^8\right) \div\left(8.3 \times 10^3\right) \\\\ & =(5.7 \div 8.3) \times\left(10^8 \div 10^3\right) \\\\ & =0.687 \times 10^5 \text { (rounded) } \\\\ & =\left(6.87 \times 10^{-1}\right) \times 10^5 \\\\ & =6.87 \times 10^4 \end{aligned}

The factory can provide pencils to 6.87 \times 10^4 schools.

## How to divide scientific notation FAQs

What is the standard form?

This is the term for scientific notation used in the United Kingdom. There are other terms used such as ‘standard index form’ or ‘scientific form’, and they all have the same meaning as scientific notation.

What are significant digits?

This indicates the desired number of digits used to express a number’s accuracy.

How do metric prefixes connect to scientific notation?

Since the system is based on powers of 10, each power of 10 has a prefix in the metric system.

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