Find \left(1.6 \times 10^7\right) + \left(3.8 \times 10^8\right ]. Express Your Answer In Scientific Notation.

Mar 12, 2025 by ADMIN 111 views

$13. Find $\left(1.6 \times 10^7\right) + \left(3.8 \times 10^8\right)$: Expressing the Answer in Scientific Notation$

===========================================================

Introduction

Scientific notation is a way of expressing very large or very small numbers in a more manageable form. It is commonly used in mathematics, physics, and engineering to simplify calculations and make it easier to understand complex concepts. In this problem, we will learn how to add two numbers expressed in scientific notation and express the answer in the same format.

Understanding Scientific Notation

Before we proceed with the problem, let's briefly review what scientific notation is. Scientific notation is a way of expressing a number as a product of a number between 1 and 10 and a power of 10. For example, the number 456,000 can be expressed in scientific notation as $4.56 \times 10^5$ . Similarly, the number 0.000456 can be expressed in scientific notation as $4.56 \times 10^{-4}$ .

Adding Numbers in Scientific Notation

To add two numbers expressed in scientific notation, we need to follow a few simple steps:

Ensure the powers of 10 are the same: If the powers of 10 are different, we need to adjust the numbers so that they have the same power of 10.
Add the coefficients: Once the powers of 10 are the same, we can add the coefficients (the numbers in front of the powers of 10).
Express the answer in scientific notation: Finally, we need to express the answer in scientific notation.

Solving the Problem

Now that we have a clear understanding of how to add numbers in scientific notation, let's solve the problem.

Step 1: Ensure the Powers of 10 are the Same

The first step is to ensure that the powers of 10 are the same. In this case, we have $1.6 \times 10^7$ and $3.8 \times 10^8$ . To make the powers of 10 the same, we need to adjust the numbers so that they have the same power of 10.

Step 2: Adjust the Numbers

To adjust the numbers, we can multiply the first number by $10^{8-7}$ , which is equal to $10^1$ . This will give us $1.6 \times 10^7 \times 10^1$ , which is equal to $1.6 \times 10^8$ .

Step 3: Add the Coefficients

Now that the powers of 10 are the same, we can add the coefficients. We have $1.6 \times 10^8$ and $3.8 \times 10^8$ . Adding the coefficients, we get $1.6 + 3.8 = 5.4$ .

Step 4: Express the Answer in Scientific Notation

Finally, we need to express the answer in scientific notation. We have $5.4 \times 10^8$ . This is already in scientific notation, so we don't need to do anything else.

Conclusion

In this problem, we learned how to add two numbers expressed in scientific notation and express the answer in the same format. We followed a few simple steps to ensure the powers of 10 were the same, added the coefficients, and expressed the answer in scientific notation. With practice, you will become more comfortable working with numbers in scientific notation and be able to solve more complex problems.

Example Use Cases

Scientific notation is used in a wide range of applications, including:

Physics and Engineering: Scientific notation is used to express large and small numbers in physics and engineering, such as distances, velocities, and forces.
Computer Science: Scientific notation is used in computer science to represent large and small numbers in algorithms and data structures.
Finance: Scientific notation is used in finance to represent large and small numbers in financial calculations, such as interest rates and investment returns.

Tips and Tricks

Here are a few tips and tricks to help you work with numbers in scientific notation:

Use a calculator: When working with large and small numbers, it's often easier to use a calculator to perform calculations.
Use exponent rules: When working with exponents, it's often easier to use exponent rules to simplify calculations.
Check your units: When working with units, it's often easier to check your units to ensure that they are correct.

Common Mistakes

Here are a few common mistakes to avoid when working with numbers in scientific notation:

Incorrect powers of 10: Make sure to get the powers of 10 correct when adding or subtracting numbers in scientific notation.
Incorrect coefficients: Make sure to get the coefficients correct when adding or subtracting numbers in scientific notation.
Incorrect units: Make sure to get the units correct when working with numbers in scientific notation.

Conclusion

In conclusion, working with numbers in scientific notation can be challenging, but with practice and patience, you will become more comfortable and confident. Remember to follow the steps outlined in this problem, and don't be afraid to ask for help if you need it. With time and practice, you will become a master of working with numbers in scientific notation.

===========================================================

Q&A: Working with Numbers in Scientific Notation

=====================================================

Q: What is scientific notation?

A: Scientific notation is a way of expressing very large or very small numbers in a more manageable form. It is commonly used in mathematics, physics, and engineering to simplify calculations and make it easier to understand complex concepts.

Q: How do I express a number in scientific notation?

A: To express a number in scientific notation, you need to write it as a product of a number between 1 and 10 and a power of 10. For example, the number 456,000 can be expressed in scientific notation as $4.56 \times 10^5$ .

Q: How do I add numbers in scientific notation?

A: To add numbers in scientific notation, you need to follow a few simple steps:

Ensure the powers of 10 are the same: If the powers of 10 are different, you need to adjust the numbers so that they have the same power of 10.
Add the coefficients: Once the powers of 10 are the same, you can add the coefficients (the numbers in front of the powers of 10).
Express the answer in scientific notation: Finally, you need to express the answer in scientific notation.

Q: What if the powers of 10 are different?

A: If the powers of 10 are different, you need to adjust the numbers so that they have the same power of 10. You can do this by multiplying the first number by $10^{8-7}$ , which is equal to $10^1$ . This will give you $1.6 \times 10^7 \times 10^1$ , which is equal to $1.6 \times 10^8$ .

Q: How do I express the answer in scientific notation?

A: Once you have added the coefficients, you need to express the answer in scientific notation. You can do this by writing the answer as a product of a number between 1 and 10 and a power of 10.

Q: What are some common mistakes to avoid when working with numbers in scientific notation?

A: Some common mistakes to avoid when working with numbers in scientific notation include:

Incorrect powers of 10: Make sure to get the powers of 10 correct when adding or subtracting numbers in scientific notation.
Incorrect coefficients: Make sure to get the coefficients correct when adding or subtracting numbers in scientific notation.
Incorrect units: Make sure to get the units correct when working with numbers in scientific notation.

Q: How do I use a calculator to work with numbers in scientific notation?

A: You can use a calculator to work with numbers in scientific notation by entering the numbers in scientific notation and using the calculator's exponent and multiplication functions to perform calculations.

Q: How do I use exponent rules to simplify calculations with numbers in scientific notation?

A: You can use exponent rules to simplify calculations with numbers in scientific notation by using the rules of exponents to combine and simplify expressions.

Q: What are some real-world applications of scientific notation?

A: Scientific notation is used in a wide range of applications, including:

Physics and Engineering: Scientific notation is used to express large and small numbers in physics and engineering, such as distances, velocities, and forces.
Computer Science: Scientific notation is used in computer science to represent large and small numbers in algorithms and data structures.
Finance: Scientific notation is used in finance to represent large and small numbers in financial calculations, such as interest rates and investment returns.