What Is The Product Of $3.78 \times 10^9$ And $7.35 \times 10^5$?A. $ 2.7783 × 10 14 2.7783 \times 10^{14} 2.7783 × 1 0 14 [/tex] B. $2.7783 \times 10^{15}$ C. $2.7783 \times 10^{45}$ D. $2.7783 \times
What is the Product of Two Large Numbers in Scientific Notation?
When dealing with large numbers in scientific notation, it's essential to understand the rules for multiplying them. In this article, we'll explore how to multiply two numbers in scientific notation and provide a step-by-step guide to help you calculate the product.
Understanding Scientific Notation
Scientific notation is a way of expressing numbers in the form a × 10^n, where a is a number between 1 and 10, and n is an integer. This notation is commonly used to represent large or small numbers in a more manageable form. For example, the number 3.78 × 10^9 can be read as "3.78 times ten to the power of 9" or "3.78 billion."
Multiplying Numbers in Scientific Notation
To multiply two numbers in scientific notation, we need to follow a specific set of rules. The rules are as follows:
- Multiply the coefficients (the numbers in front of the powers of 10).
- Add the exponents (the powers of 10).
- Write the product in scientific notation.
Let's apply these rules to the given problem:
Step 1: Multiply the Coefficients
The coefficients are 3.78 and 7.35. To multiply them, we simply multiply the numbers:
3.78 × 7.35 = 27.783
Step 2: Add the Exponents
The exponents are 9 and 5. To add them, we simply add the numbers:
9 + 5 = 14
Step 3: Write the Product in Scientific Notation
Now that we have the product of the coefficients and the sum of the exponents, we can write the product in scientific notation:
However, we need to express the coefficient in the form a × 10^b, where a is a number between 1 and 10. To do this, we can rewrite the coefficient as:
Using the rule that a × 10^b × 10^c = a × 10^(b+c), we can simplify the expression:
Therefore, the product of $3.78 \times 10^9$ and $7.35 \times 10^5$ is:
Conclusion
In this article, we explored how to multiply two numbers in scientific notation. We followed the rules for multiplying coefficients and adding exponents, and we expressed the product in scientific notation. The product of $3.78 \times 10^9$ and $7.35 \times 10^5$ is $2.7783 \times 10^{14}$.
Frequently Asked Questions (FAQs) About Multiplying Numbers in Scientific Notation
In the previous article, we explored how to multiply two numbers in scientific notation. However, we understand that you may still have some questions about this topic. In this article, we'll address some of the most frequently asked questions about multiplying numbers in scientific notation.
Q: What is the rule for multiplying numbers in scientific notation?
A: The rule for multiplying numbers in scientific notation is as follows:
- Multiply the coefficients (the numbers in front of the powers of 10).
- Add the exponents (the powers of 10).
- Write the product in scientific notation.
Q: How do I multiply numbers with different powers of 10?
A: When multiplying numbers with different powers of 10, you need to add the exponents. For example, if you have 3.78 × 10^9 and 7.35 × 10^5, you would add the exponents 9 and 5 to get 14.
Q: Can I multiply numbers in scientific notation using a calculator?
A: Yes, you can multiply numbers in scientific notation using a calculator. Most calculators have a scientific notation mode that allows you to enter numbers in this format. Simply enter the numbers and the calculator will perform the multiplication and display the result in scientific notation.
Q: How do I convert a number from standard notation to scientific notation?
A: To convert a number from standard notation to scientific notation, you need to move the decimal point to the left or right until you have a number between 1 and 10. Then, you need to multiply the number by 10 raised to the power of the number of places you moved the decimal point.
Q: Can I multiply numbers in scientific notation with negative exponents?
A: Yes, you can multiply numbers in scientific notation with negative exponents. When multiplying numbers with negative exponents, you need to follow the same rules as before, but you need to be careful with the signs. For example, if you have 3.78 × 10^-9 and 7.35 × 10^5, you would multiply the coefficients and add the exponents, but you would need to be careful with the signs.
Q: How do I simplify a product in scientific notation?
A: To simplify a product in scientific notation, you need to combine the coefficients and the exponents. If the coefficients are in the form a × 10^b and a × 10^c, you can combine them by multiplying the coefficients and adding the exponents. For example, if you have 2.7783 × 10^14 and 3.14159 × 10^14, you can combine them by multiplying the coefficients and adding the exponents.
Q: Can I multiply numbers in scientific notation with decimal points?
A: Yes, you can multiply numbers in scientific notation with decimal points. When multiplying numbers with decimal points, you need to follow the same rules as before, but you need to be careful with the decimal points. For example, if you have 3.78 × 10^9 and 7.35 × 10^5, you would multiply the coefficients and add the exponents, but you would need to be careful with the decimal points.
Conclusion
In this article, we addressed some of the most frequently asked questions about multiplying numbers in scientific notation. We hope that this article has provided you with a better understanding of this topic and has helped you to feel more confident when working with numbers in scientific notation.