# Multiplying Scientific Notation Worksheet, Examples, and Rules

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### Key Points about Multiplying in Scientific Notation

- Scientific notation is a way of expressing numbers as the product of a decimal number and a power of 10.
- To multiply scientific notation, one needs to follow a set of rules that involve multiplying the decimal numbers and adding the exponents.
- Multiplying scientific notation is a fundamental concept in mathematics that is essential for many fields, including physics, chemistry, and engineering.

## Multiplying in Scientific Notation

When **Multiplying in Scientific Notation** you multiply the coefficients in the problem. **Multiplying in Scientific Notation** is different then Adding or Subtracting because you do not have to make the exponents equal beforehand. When multiplying the coefficients, you will add the exponents together. After you multiply check your answer to make sure the coefficient is in between 1 and 10. If the coefficient is not in between 1 and 10, you must move the decimal to make it in between 1 and 10. You add to the exponent for each space that you moved the decimal to the left. You subtract from the exponent for each space that you moved the decimal to the right.

Multiplying scientific notation is a fundamental concept in mathematics that is used to simplify large or small numbers. Scientific notation is a way of expressing numbers as the product of a decimal number and a power of 10. It is useful for representing very large or very small numbers that are difficult to read or write in standard decimal notation.

To multiply scientific notation, one needs to follow a set of rules that involve multiplying the decimal numbers and adding the exponents. The process can be broken down into three steps: multiply the decimal numbers, add the exponents, and express the result in proper scientific notation. It is important to note that the rules for multiplying scientific notation apply to both positive and negative exponents.

Multiplying scientific notation is a skill that is essential for many fields, including physics, chemistry, and engineering. It allows scientists and engineers to work with very large or very small numbers in a more manageable way. Understanding how to multiply scientific notation is a key step in mastering the basics of algebra and calculus.

**Common Core Standard: **8.EE.A.3**Related Topics: **Square Roots, Cube Roots, Irrational Numbers, Powers of 10, Scientific Notation Intro, Converting Numbers to Scientific Notation, Converting Numbers from Scientific Notation, Adding and Subtracting in Scientific Notation, Dividing in Scientific Notation**Return To: **Home, 8th Grade

## What is Scientific Notation?

Scientific notation is a way of representing numbers that are either very large or very small. It is a shorthand method that uses exponents to express a number as a coefficient multiplied by 10 raised to a power. The coefficient is a number between 1 and 10, and the exponent is an integer.

In scientific notation, numbers are expressed in the form of a × 10^n, where “a” is the decimal number between 1 and 10, and “n” is the exponent that represents the number of places the decimal point must be moved to obtain the original number. For example, the number 3,000,000 can be expressed as 3 × 10^6 in scientific notation.

Scientific notation is useful because it simplifies calculations with very large or small numbers. It also makes it easier to compare and analyze data, particularly when dealing with large datasets.

To convert a number to scientific notation, the decimal point is moved to the left or right so that only one digit remains to the left of the decimal point. The number of places the decimal point is moved becomes the exponent. If the decimal point is moved to the left, the exponent is positive; if it is moved to the right, the exponent is negative.

For example, the number 0.000000123 can be expressed in scientific notation as 1.23 × 10^-7. The coefficient is 1.23, which is the decimal number between 1 and 10, and the exponent is -7, which represents the number of places the decimal point has been moved to the right.

In summary, scientific notation is a way of expressing numbers that are very large or very small using exponents. It is a useful shorthand method that simplifies calculations and makes it easier to compare and analyze data.

## How to Multiply Scientific Notation

Multiplying numbers in scientific notation involves multiplying their coefficients and adding their exponents. Here are the steps to follow:

Multiply the coefficients: To multiply two numbers in scientific notation, the first step is to multiply their coefficients. For example, to multiply 2.5 x 10^3 and 3.2 x 10^4, you would multiply 2.5 and 3.2 to get 8.0.

Add the exponents: The second step is to add the exponents. In the example above, the exponents are 3 and 4, so you would add them to get 7.

Express the result in scientific notation: The final step is to express the result in scientific notation. In the example above, the result is 8.0 x 10^7.

It’s important to note that if the product of the coefficients is not a number between 1 and 10, you will need to adjust the exponent accordingly. For example, if the product of the coefficients is 80, you would need to adjust the exponent to 8 x 10^8.

Calculators can be helpful when multiplying numbers in scientific notation, as they can handle the arithmetic and exponentiation automatically. There are many online calculators available that can help with this task.

Here are a few examples of how to multiply numbers in scientific notation:

Example 1: (2.5 x 10^3) x (3.2 x 10^4)

- Multiply the coefficients: 2.5 x 3.2 = 8.0
- Add the exponents: 3 + 4 = 7
- Express the result in scientific notation: 8.0 x 10^7

Example 2: (6.7 x 10^-2) x (2.1 x 10^5)

- Multiply the coefficients: 6.7 x 2.1 = 14.07
- Add the exponents: -2 + 5 = 3
- Express the result in scientific notation: 1.407 x 10^3

By following these steps, anyone can multiply numbers in scientific notation with ease.

## How to Multiply Scientific Notation with Different Exponents

Multiplying scientific notation with different exponents can seem daunting, but it is actually quite simple. The key is to remember the rules of exponents and to regroup the factors of 10 to simplify the expression. Here are the steps to follow:

Multiply the coefficients: The first step is to multiply the coefficients of the two numbers in scientific notation. For example, to multiply 3.2 x 10^4 and 6.5 x 10^2, you would multiply 3.2 and 6.5 to get 20.8.

Multiply the powers of 10: The second step is to multiply the powers of 10. To do this, add the exponents of the two numbers. For example, to multiply 3.2 x 10^4 and 6.5 x 10^2, you would add 4 and 2 to get 6. The result is 20.8 x 10^6.

Regroup the factors of 10: The final step is to regroup the factors of 10 to simplify the expression. In the example above, 20.8 x 10^6 can be simplified to 2.08 x 10^7 by regrouping the factor of 10.

It is important to note that when multiplying scientific notation with different exponents, the resulting exponent will be positive if both exponents are positive, and negative if one of the exponents is negative.

In summary, to multiply scientific notation with different exponents, multiply the coefficients, add the exponents, and regroup the factors of 10. By following these simple steps, anyone can easily multiply scientific notation with different exponents.

## How to Multiply Scientific Notation with Negative Exponents

When multiplying scientific notation with negative exponents, it is important to understand the basic rules of exponents and how they apply to scientific notation. Scientific notation is a way of expressing numbers in the form of a coefficient multiplied by a power of 10. Negative exponents indicate that the number is less than one and needs to be divided by a power of 10.

To multiply scientific notation with negative exponents, follow these steps:

Multiply the coefficients: The first step is to multiply the coefficients of the two numbers. For example, if you are multiplying 2.5 x 10^-3 and 3.0 x 10^-4, you would multiply 2.5 and 3.0 to get 7.5.

Add the exponents: The second step is to add the exponents of the two numbers. In our example, the exponents are -3 and -4, so we would add them to get -7.

Rewrite the answer in scientific notation: The final step is to rewrite the answer in scientific notation. In our example, the answer is 7.5 x 10^-7.

It is important to note that if the answer is not in scientific notation, you may need to regroup a factor of 10. For example, if the answer to the above problem was 0.00000075, you would need to regroup the answer as 7.5 x 10^-7.

In summary, to multiply scientific notation with negative exponents, multiply the coefficients, add the exponents, and rewrite the answer in scientific notation.

## 3 Simple Multiplying Scientific Notation Examples

Multiplying scientific notation is a fundamental skill that is widely used in various fields such as engineering, physics, and chemistry. It involves multiplying the coefficients and adding the exponents of the powers of 10.

- Multiply the coefficients together.
- Add the exponents of the power of ten together.
- If the coefficient is in between 1 and 10 then you are done solving.
- If the coefficient is not in between 1 and 10, then you must move the decimal either left or right to make the coefficient in between 1 and 10.
- If you move the decimal left, you add to the exponent, if you move the decimal right, you subtract from the exponent.

Here are a few examples that demonstrate how to multiply scientific notation:

### Example 1:

Multiply (2.4 x 10^5) and (3.2 x 10^3)

- Multiply the coefficients: 2.4 x 3.2 = 7.68
- Add the exponents: 10^5 x 10^3 = 10^8
- Combine the coefficient and the exponent: 7.68 x 10^8

Therefore, (2.4 x 10^5) x (3.2 x 10^3) = 7.68 x 10^8

### Example 2:

Multiply (5.6 x 10^-3) and (2.5 x 10^-2)

- Multiply the coefficients: 5.6 x 2.5 = 14.0
- Add the exponents: 10^-3 + 10^-2 = 10^-5
- Combine the coefficient and the exponent: 14.0 x 10^-5

Therefore, (5.6 x 10^-3) x (2.5 x 10^-2) = 14.0 x 10^-5

### Example 3:

Multiply (6.4 x 10^4) and (7.5 x 10^-3)

- Multiply the coefficients: 6.4 x 7.5 = 48.0
- Add the exponents: 10^4 x 10^-3 = 10^1
- Combine the coefficient and the exponent: 48.0 x 10^1

Therefore, (6.4 x 10^4) x (7.5 x 10^-3) = 48.0 x 10^1

These examples demonstrate how to multiply scientific notation by following the basic rules of multiplying coefficients and adding exponents. It is important to note that the result should always be expressed in proper scientific notation.

For more complex problems, one can use a multiplying scientific notation calculator to solve them quickly and accurately.

## 5 Quick Multiplying Scientific Notation Problems

## Multiplying Scientific Notation Rules

Multiplying numbers in scientific notation involves multiplying the decimal numbers and adding the exponents. Here are the rules to follow when multiplying numbers in scientific notation:

Multiply the decimal numbers: When multiplying two numbers in scientific notation, multiply the decimal numbers together. For example, when multiplying 3.2 x 10^4 and 5.6 x 10^3, multiply 3.2 and 5.6 to get 17.92.

Add the exponents: After multiplying the decimal numbers, add the exponents of the two numbers. In the example above, add the exponents 4 and 3 to get 7.

Express the results in scientific notation: Finally, express the result in scientific notation by writing the decimal number and the exponent. In the example above, the result is 1.792 x 10^7.

It is important to note that when multiplying numbers in scientific notation, the exponent of the result is the sum of the exponents of the two numbers being multiplied. If the result is not in scientific notation, it can be converted by regrouping a factor of 10.

Multiplying numbers with positive and negative exponents follows the same rules as multiplying positive exponents. When multiplying numbers with negative exponents, the negative sign is treated as part of the exponent. For example, when multiplying 3.2 x 10^4 and 5.6 x 10^-3, multiply 3.2 and 5.6 to get 17.92 and add the exponents 4 and -3 to get 1.792 x 10^1.

In summary, multiplying numbers in scientific notation involves multiplying decimal numbers and adding exponents. The result is expressed in scientific notation by writing the decimal number and the exponent. The same rules apply when multiplying numbers with positive and negative exponents.

## Multiplying Scientific Notation FAQ

### How do you multiply numbers in scientific notation with negative exponents?

To multiply numbers in scientific notation with negative exponents, one can follow these steps:

- Multiply the coefficients (the numbers in front of the powers of 10).
- Add the exponents of 10.
- If the result is not in scientific notation, convert it to scientific notation.

### How can you multiply scientific notation by a whole number?

To multiply scientific notation by a whole number, one can follow these steps:

- Rewrite the whole number as a power of 10.
- Multiply the coefficients (the numbers in front of the powers of 10).
- Add the exponents of 10.
- If the result is not in scientific notation, convert it to scientific notation.

### What is the process for dividing numbers in scientific notation with different exponents?

To divide numbers in scientific notation with different exponents, one can follow these steps:

- Divide the coefficients (the numbers in front of the powers of 10).
- Subtract the exponent of the divisor from the exponent of the dividend.
- If the result is not in scientific notation, convert it to scientific notation.

### How do you add numbers in scientific notation?

To add numbers in scientific notation, one can follow these steps:

- Rewrite the numbers with the same exponent of 10.
- Add the coefficients (the numbers in front of the powers of 10).
- Keep the exponent of 10 the same.

### What are the steps for multiplying and dividing numbers in scientific notation?

To multiply numbers in scientific notation, one can follow these steps:

- Multiply the coefficients (the numbers in front of the powers of 10).
- Add the exponents of 10.

To divide numbers in scientific notation, one can follow these steps:

- Divide the coefficients (the numbers in front of the powers of 10).
- Subtract the exponent of the divisor from the exponent of the dividend.

### What are some practice problems for multiplying and dividing numbers in scientific notation?

Practice problems for multiplying and dividing numbers in scientific notation can be found in textbooks, online resources, and educational videos such as those provided by Khan Academy.

### What are the rules for scientific notation?

The rules for scientific notation are:

- The coefficient must be between 1 and 10 (inclusive).
- The exponent of 10 must be an integer.
- The number can be positive or negative.

### How do you multiply and divide exponents?

To multiply exponents, one can add the exponents if the bases are the same.

To divide exponents, one can subtract the exponent of the divisor from the exponent of the dividend if the bases are the same.

## Multiplying Scientific Notation Worksheet Video Explanation

Watch our free video on how to solve **Multiplying in Scientific Notation**. This video shows how to solve problems that are on our free** Multiply Scientific Notation **worksheet that you can get by submitting your email above.

**Watch the free Multiply Scientific Notation video on YouTube here:** **How to Multiply Scientific Notation**

**Video Transcript:**

This video is about how to multiply scientific notation. You can get the scientific notation multiplication worksheet we use in this video for free by clicking on the link in the description below.

Here we are at our first problem for scientific notation multiplication and we have to multiply our two numbers written in scientific notation together. When multiplying numbers in scientific notation you are going to multiply the two coefficients together. In this case our coefficients are nine and two. We will multiply these together and then you will add the exponents of three and five, just like if you were to follow the rule for multiplying exponents. If we wanted to multiply 10 to the third times 10 to the fifth, we would keep the base of 10 and then we would add three plus five, which would equal 10 to the 8th. You will follow the same rule when multiplying numbers in scientific notation and finding out how do you multiply scientific notation. Let’s go ahead and solve our first problem.

9 times 10 to the 3rd times 2 times 10 to the 5th. Like we already said we have to multiply the coefficients together. We’re going to do 9 times 2 because we’re multiplying and that’s gonna be times and then we have our 2 powers of 10 which are 10 to the third and 10 to the fifth. That will be times 10 to the 3 plus 5 because we have to add the exponents 9 times 2 is 18 times 10 to the 3 plus 5 which is 8. Now 18 times 10 to the 8th is the result of multiplying these together, however, 18 is not in scientific notation. This coefficient here has to be in between 1 and 10.

What we’re going to do is we’re going to take our decimal we’re gonna move it to the left once. And then we’re going to add 1 to the exponent. Every time you move the decimal to the left you will add to the exponent. For example, if we had to move the decimal 2 times we would add two to the exponent but in this case we only have to move the decimal one time to make a number in between one and ten. We only add one to the exponent, so after we move the decimal one time it becomes 1.8 times 10 to the 8 plus 1, which is ninth power. Our final solution is 1.8 times 10 to the ninth power.

Number 4 on the multiplying with scientific notation worksheet gives us 4 point 2 times 10 to the first times 6 point 5 times 10 to the ninth. This problem shows how to multiply scientific notation with different exponents. We already know that when we multiply in scientific notation, we’re going to take our coefficients and we’re going to multiply them together. So we’re going to do four point two times six point five times our power of 10, which is 10 to the 1st plus 9 because this one’s 10 to the ninth.

We will do four point two times six point five which is twenty seven point three times 10 to the 1 plus nine which is 10. Now once again our coefficient here has to be in between 1 and 10 so we have to take our decimal and moved it left once so that it becomes two point seven three which is in between 1 and 10. And then add one to the exponent because we move the decimal to the left so this will become two point seven three times 10 to the 10 plus one power, which is to the 11th and this is going to be our final solution. This is how to multiply numbers in scientific notation.

The last problem we’re going to do to show you how to multiply numbers written in scientific notation is number five. This problem gives us 3 times 10 to the 15th times 8 times 10 to the 12th. Once again we will multiply our coefficients together. We’ll do 3 times 8 times our power of 10, which in this case is 10 to the 15th and 10 to the 12th. 10 to the 15 plus 12 3 times 8 is 24 times 10. And then our power is going to be 15 plus 12 which is 27.

Then our coefficient here is 24 that is not in between ten so we have to move the decimal from 24 to 2.4. That it’s in between 1 and 10 and then because we move the decimal left once we will add 1 to our exponent so this will become 2 point 4 times 10 to the 28th power, which is our answer. You can try all the practice problems by downloading the free multiplying with scientific notation worksheet above.

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