# Using Operations with Scientific Notations

## Operations with Scientific Notation

In this text, we're going to explore how to perform operations like multiplication and division with numbers expressed in scientific notation. This skill is essential for handling very large or small numbers efficiently in math and science. We'll also touch on how to convert numbers to scientific notation, providing a comprehensive understanding of this useful mathematical tool.

For a refresher on the scientific notation rules, check out this video: Reading and Writing Scientific Notation

## Understanding Operations with Scientific Notation

Scientific Notation is a way of expressing numbers that are very large or very small. It's expressed as a product of a number between 1 and 10 and a power of ten and can look like this: $4.3 \times 10^2$.

In the following section, you will learn the steps to finding the product or quotient of numbers in scientific notation.

### Multiplying Numbers in Scientific Notation – Steps

• Multiply the decimal parts.
• Add the exponents of 10.

### Dividing Numbers in Scientific Notation – Steps

• Divide the decimal parts.
• Subtract the exponent of the divisor from the exponent of the dividend.

Let’s check your understanding so far.

### Adjusting the Solution to Scientific Notation

Don’t forget, according to the scientific notation definition, the decimal part should be greater than or equal to 1 and less than 10. Adjusting the decimal and the exponent accordingly will ensure your result is in the correct form of scientific notation.

Decimal is greater than or equal to 10 Shift decimal left; Increase exponent $10 × 10^5$ $1.0 × 10^6$
Decimal is less than 1 Shift decimal right; Decrease exponent $0.4 × 10^3$ $4.0 × 10^2$

Let’s check your understanding so far:

The number $42.5 \times 10^3$ is not in scientific notation. How can it be re-written to be in scientific notation?
The number $0.091 \times 10^5$ is not in scientific notation. How can it be re-written to be in scientific notation?

## Operations with Scientific Notation – Guided Practice

Let’s walk through some examples of multiplying and dividing numbers in scientific notation:

Divide $8 \times 10^6$ by $4 \times 10^2$.
Multiply $3.5 \times 10^2$ by $2 \times 10^3$.

## Operations with Scientific Notation – Real-World Application

Now that you have learned how to multiply and divide numbers written in scientific notation, let’s try to complete a few real-world problems.

## Operations with Scientific Notation – Summary

Key Points from This Text:

• Scientific notation simplifies the process of working with very large or small numbers.
• Multiplication involves multiplying the decimal parts and adding the exponents.
• Division includes dividing the decimal parts and subtracting the exponents.
• Converting to and from scientific notation is an essential skill in these operations.

Explore more topics and practice problems on our website, including interactive exercises, videos, and worksheets to further enhance your understanding of scientific notation and other math concepts!

If you are looking for information on adding numbers in scientific notation or subtracting numbers in scientific notation, this video is a great resource: Adding and Subtracting Numbers in Scientific Notation

## Operations with Scientific Notation – Frequently Asked Questions

Why is scientific notation used in real-world scenarios?
Can scientific notation be used for both very large and very small numbers?
How does multiplying in scientific notation differ from standard multiplication?
Is there a quick way to check if my scientific notation result is correct?
When dividing in scientific notation, why do we subtract the exponents?
How important is it to write numbers in scientific notation in their proper form?
Can I calculate scientific notation on a calculator for complex problems?
Are there any real-world professions that frequently use scientific notation?
How does understanding scientific notation benefit students in their studies?
What are some common errors to avoid when working with scientific notation?
Can I practice operations with scientific notation using everyday numbers?

## Using Operations with Scientific Notations exercise

Would you like to apply the knowledge you’ve learned? You can review and practice it with the tasks for the learning text Using Operations with Scientific Notations.
• ### Calculate the standard form multiplication.

Hints

To make it easier rearrange it.

$2\times4$ and $10^{3}\times10^{5}$

When multiplying indices with the same base we add the powers.

$2\times4 = 8$

$8\times(10^{3}\times10^{5})$

Solution

$8\times10^{8}$

• Multiply $2$ and $4$ together
• Add the powers $3$ and $5$ = $10^{8}$
• ### Calculate the standard form division

Hints

To make it easier rearrange it.

$9\div3$ and $10^{7}\div10^{5}$

When dividing indices with the same base we subtract the powers.

$9\div3 = 3$

$3\times(10^{7}\div10^{5})$

Solution

$3\times10^{2}$

• Divide $9$ by $3$
• Subtract the powers $7$ and $5$ = $10^{2}$
• ### Find the product of two values in standard form

Hints

To make it easier rearrange it.

$4\times6$ and $10^{8}\times10^{4}$

When multiplying indices with the same base we ADD the powers.

$4\times6 = 24$

$24\times(10^{8}\times10^{4}) = 24\times10^{12}$ (We add the powers here)

The answer is not in standard form.

For this $24$ needs to be a number between $1$ and $10$ but this means we have to add one more to the indices.

Solution

$2.4\times10^{13}$ in standard form.

• ### Calculate and leave the answer in standard form.

Hints

Some of the answers will need putting into standard form when the calculation has been done.

For example, if the answer is $27\times10^{3}$, we need to write it as a number between $1$ and $10$ and make adjustments.

$2.7\times10^{4}$

If the answer is already a number between $1$ and $10$ then no adjustment is necessary it is in standard form.

Solution
• $(45\times10^{6})\div(3\times10^{2}) = 1.5\times10^{5}$
You can see this explained in detail above.

• $(7\times10^{3})\times(6\times10^{2}) = 4.2\times10^{6}$
• $(6\times10^{4})\times(7\times10^{2}) = 4.2\times10^{7}$
• $(42\times10^{8})\div(6\times10^{2}) = 7\times10^{6}$
• ### Multiplying and dividing by powers of ten.

Hints

When multiplying indices we add the powers if the base is the same.

When dividing indices we subtract the powers if the base is the same.

Solution

• $10^{6}\times10^{2} = 10^{8}$
• $10^{3}\times10^{4} = 10^{7}$
• $10^{8}\div10^{2} = 10^{6}$
• $10^{5}\div10^{2} = 10^{3}$
When multiplying indices we add the powers if the base is the same.

When dividing indices we subtract the powers if the base is the same.

• ### Divide the standard form numbers.

Hints

The operation we use for this problem is division.

Distance from London to Sydney $\div$ Distance from London to Paris

Distance from London to Sydney $\div$ Distance from London to Paris

$(1.05\times10^{4})\div(2.9\times10^{2})$

When we have divided $(1.05\times10^{4})\div(2.9\times10^{2})$ we need to make an adjustment to get the answer to the nearest whole number.

Solution

Answer is $36$

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The authors
sofatutor Team
Using Operations with Scientific Notations
8.EE.A.4