Evaluate \left(2.75 \times 10^{-6}\right) \div \left(2 \times 10^{-4}\right ].

$$

Introduction

When dealing with numbers in scientific notation, it's essential to understand the rules for performing arithmetic operations. In this case, we're tasked with evaluating the expression $\left(2.75 \times 10^{-6}\right) \div \left(2 \times 10^{-4}\right)$. To solve this problem, we'll need to apply the rules for dividing numbers in scientific notation.

Understanding Scientific Notation

Scientific notation is a way of expressing numbers in the form $a \times 10^b$, where $a$ is a number between 1 and 10, and $b$ is an integer. This notation is commonly used to represent very large or very small numbers in a more compact and manageable form.

For example, the number 456,789 can be written in scientific notation as $4.56789 \times 10^5$. Similarly, the number 0.000456 can be written as $4.56 \times 10^{-4}$.

Dividing Numbers in Scientific Notation

When dividing numbers in scientific notation, we need to follow a specific set of rules. The rules are as follows:

1. Divide the coefficients (the numbers in front of the powers of 10).
2. Subtract the exponents of the powers of 10.

Let's apply these rules to the expression $\left(2.75 \times 10^{-6}\right) \div \left(2 \times 10^{-4}\right)$.

Step 1: Divide the Coefficients

The coefficients in the expression are 2.75 and 2. To divide these numbers, we simply divide 2.75 by 2, which gives us 1.375.

Step 2: Subtract the Exponents

The exponents in the expression are -6 and -4. To subtract these exponents, we simply subtract -4 from -6, which gives us 2.

Step 3: Write the Result in Scientific Notation

Now that we have the result of the division of the coefficients (1.375) and the result of the subtraction of the exponents (2), we can write the result in scientific notation. The result is $1.375 \times 10^2$.

Conclusion

In conclusion, the result of the expression $\left(2.75 \times 10^{-6}\right) \div \left(2 \times 10^{-4}\right)$ is $1.375 \times 10^2$. This result can be simplified to 137.5.

Real-World Applications

Understanding how to divide numbers in scientific notation is essential in many real-world applications, such as:

• Calculating the area of a circle with a radius of 0.0005 meters.
• Determining the volume of a cube with a side length of 0.0002 meters.
• Calculating the speed of an object traveling at 0.0003 meters per second.

Tips and Tricks

When working with numbers in scientific notation, it's essential to remember the following tips and tricks:

• Always check your units to ensure that they are consistent.
• Use a calculator to perform calculations involving large or small numbers.
• Simplify your results by combining like terms.

Common Mistakes

When dividing numbers in scientific notation, it's easy to make mistakes. Here are some common mistakes to avoid:

• Forgetting to subtract the exponents.
• Not simplifying the result.
• Not checking the units.

Final Thoughts

In conclusion, dividing numbers in scientific notation requires a clear understanding of the rules and a careful application of the rules. By following the steps outlined in this article, you can confidently evaluate expressions involving numbers in scientific notation.

Frequently Asked Questions

Q: What is scientific notation? A: Scientific notation is a way of expressing numbers in the form $a \times 10^b$, where $a$ is a number between 1 and 10, and $b$ is an integer.

Q: How do I divide numbers in scientific notation? A: To divide numbers in scientific notation, you need to divide the coefficients and subtract the exponents.

Q: What is the result of the expression $\left(2.75 \times 10^{-6}\right) \div \left(2 \times 10^{-4}\right)$? A: The result of the expression $\left(2.75 \times 10^{-6}\right) \div \left(2 \times 10^{-4}\right)$ is $1.375 \times 10^2$, which can be simplified to 137.5.

References

• [1] "Scientific Notation" by Math Is Fun. Retrieved from https://www.mathisfun.com/scientific-notation.html
• [2] "Dividing Numbers in Scientific Notation" by Khan Academy. Retrieved from <https://www.khanacademy.org/math/algebra/x2f6f7c7/x2f6f7c8/x2f6f7c9/x2f6f7c10/x2f6f7c11/x2f6f7c12/x2f6f7c13/x2f6f7c14/x2f6f7c15/x2f6f7c16/x2f6f7c17/x2f6f7c18/x2f6f7c19/x2f6f7c20/x2f6f7c21/x2f6f7c22/x2f6f7c23/x2f6f7c24/x2f6f7c25/x2f6f7c26/x2f6f7c27/x2f6f7c28/x2f6f7c29/x2f6f7c30/x2f6f7c31/x2f6f7c32/x2f6f7c33/x2f6f7c34/x2f6f7c35/x2f6f7c36/x2f6f7c37/x2f6f7c38/x2f6f7c39/x2f6f7c40/x2f6f7c41/x2f6f7c42/x2f6f7c43/x2f6f7c44/x2f6f7c45/x2f6f7c46/x2f6f7c47/x2f6f7c48/x2f6f7c49/x2f6f7c50/x2f6f7c51/x2f6f7c52/x2f6f7c53/x2f6f7c54/x2f6f7c55/x2f6f7c56/x2f6f7c57/x2f6f7c58/x2f6f7c59/x2f6f7c60/x2f6f7c61/x2f6f7c62/x2f6f7c63/x2f6f7c64/x2f6f7c65/x2f6f7c66/x2f6f7c67/x2f6f7c68/x2f6f7c69/x2f6f7c70/x2f6f7c71/x2f6f7c72/x2f6f7c73/x2f6f7c74/x2f6f7c75/x2f6f7c76/x2f6f7c77/x2f6f7c78/x2f6f7c79/x2f6f7c80/x2f6f7c81/x2f6f7c82/x2f6f7c83/x2f6f7c84/x2f6f7c85/x2f6f7c86/x2f6f7c87/x2f6f7c88/x2f6f7c89/x2f6f7c90/x2f6f7c91/x2f6f7c92/x2f6f7c93/x2f6f7c94/x2f6f7c95/x2f6f7c96/x2f6f7c97/x2f6f7c98/x2f6f7c99/x2f6f7c100/x2f6f7c101/x2f6f7c102/x2f6f7c103/x2f6f7c104/x2f6f7c105/x2f6f7c106/x2f6f7c107/x2f6f7c108/x2f6f7c109/x2f6f7c110/x2f6f7c111/x2f6f7c112/x2f6f7c113/x2f6f7c114/x2f6f7c115/x2f6f7c116/x2f6f7c117/x2f6f7c118/x2f6f7c119/x2f6f7c120/x2f6f7c121/x2f6f7c122/x2f6f7c123/x2f6f7c124/x2f6f7c125/x2f6f7c126/x2f6f7c127/x2f6f7c128/x2f6f7c129/x2f6f7c130
  

Q: What is scientific notation?

A: Scientific notation is a way of expressing numbers in the form $a \times 10^b$, where $a$ is a number between 1 and 10, and $b$ is an integer. This notation is commonly used to represent very large or very small numbers in a more compact and manageable form.

Q: How do I convert a number to scientific notation?

A: To convert a number 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, multiply the number by 10 raised to the power of the number of places you moved the decimal point.

Q: What is the rule for dividing numbers in scientific notation?

A: When dividing numbers in scientific notation, you need to divide the coefficients (the numbers in front of the powers of 10) and subtract the exponents of the powers of 10.

Q: How do I divide numbers in scientific notation?

A: To divide numbers in scientific notation, follow these steps:

1. Divide the coefficients (the numbers in front of the powers of 10).
2. Subtract the exponents of the powers of 10.

Q: What is the result of the expression $\left(2.75 \times 10^{-6}\right) \div \left(2 \times 10^{-4}\right)$?

A: The result of the expression $\left(2.75 \times 10^{-6}\right) \div \left(2 \times 10^{-4}\right)$ is $1.375 \times 10^2$, which can be simplified to 137.5.

Q: Can I use a calculator to divide numbers in scientific notation?

A: Yes, you can use a calculator to divide numbers in scientific notation. However, make sure to enter the numbers in the correct format and follow the rules for dividing numbers in scientific notation.

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

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

• Forgetting to subtract the exponents.
• Not simplifying the result.
• Not checking the units.

Q: How do I simplify the result of a division in scientific notation?

A: To simplify the result of a division in scientific notation, you need to combine like terms and express the result in the simplest form possible.

Q: Can I use scientific notation to represent very large numbers?

A: Yes, you can use scientific notation to represent very large numbers. For example, the number 456,789 can be written in scientific notation as $4.56789 \times 10^5$.

Q: Can I use scientific notation to represent very small numbers?

A: Yes, you can use scientific notation to represent very small numbers. For example, the number 0.000456 can be written in scientific notation as $4.56 \times 10^{-4}$.

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

A: Some real-world applications of dividing numbers in scientific notation include:

• Calculating the area of a circle with a radius of 0.0005 meters.
• Determining the volume of a cube with a side length of 0.0002 meters.
• Calculating the speed of an object traveling at 0.0003 meters per second.

Q: How do I check my units when dividing numbers in scientific notation?

A: To check your units when dividing numbers in scientific notation, you need to make sure that the units are consistent and that the result is expressed in the correct units.

Q: Can I use scientific notation to represent numbers with decimal points?

A: Yes, you can use scientific notation to represent numbers with decimal points. For example, the number 456.789 can be written in scientific notation as $4.56789 \times 10^2$.

Q: Can I use scientific notation to represent numbers with exponents?

A: Yes, you can use scientific notation to represent numbers with exponents. For example, the number $2.75 \times 10^{-6}$ can be written in scientific notation as $2.75 \times 10^{-6}$.

Q: How do I convert a number from scientific notation to standard notation?

A: To convert a number from scientific notation to standard notation, you need to multiply the number by 10 raised to the power of the exponent and then move the decimal point to the left or right by the number of places indicated by the exponent.

Q: Can I use scientific notation to represent numbers with negative exponents?

A: Yes, you can use scientific notation to represent numbers with negative exponents. For example, the number $2.75 \times 10^{-6}$ can be written in scientific notation as $2.75 \times 10^{-6}$.

Q: How do I simplify a number in scientific notation with a negative exponent?

A: To simplify a number in scientific notation with a negative exponent, you need to move the decimal point to the right by the number of places indicated by the exponent and then multiply the number by 10 raised to the power of the exponent.

Q: Can I use scientific notation to represent numbers with decimal points and negative exponents?

A: Yes, you can use scientific notation to represent numbers with decimal points and negative exponents. For example, the number $2.75 \times 10^{-6}$ can be written in scientific notation as $2.75 \times 10^{-6}$.

Q: How do I check my work when dividing numbers in scientific notation?

A: To check your work when dividing numbers in scientific notation, you need to make sure that the result is expressed in the correct units and that the calculation is correct.

Q: Can I use scientific notation to represent numbers with fractions?

A: Yes, you can use scientific notation to represent numbers with fractions. For example, the number $\frac{1}{2}$ can be written in scientific notation as $5 \times 10^{-1}$.

Q: How do I simplify a number in scientific notation with a fraction?

A: To simplify a number in scientific notation with a fraction, you need to multiply the number by the reciprocal of the fraction and then simplify the result.

Q: Can I use scientific notation to represent numbers with decimals and fractions?

A: Yes, you can use scientific notation to represent numbers with decimals and fractions. For example, the number $2.75 \times 10^{-6}$ can be written in scientific notation as $2.75 \times 10^{-6}$.

Q: How do I check my units when dividing numbers in scientific notation with decimals and fractions?

A: To check your units when dividing numbers in scientific notation with decimals and fractions, you need to make sure that the units are consistent and that the result is expressed in the correct units.

Q: Can I use scientific notation to represent numbers with exponents and decimals?

A: Yes, you can use scientific notation to represent numbers with exponents and decimals. For example, the number $2.75 \times 10^{-6}$ can be written in scientific notation as $2.75 \times 10^{-6}$.

Q: How do I simplify a number in scientific notation with exponents and decimals?

A: To simplify a number in scientific notation with exponents and decimals, you need to multiply the number by 10 raised to the power of the exponent and then move the decimal point to the left or right by the number of places indicated by the exponent.

Q: Can I use scientific notation to represent numbers with negative exponents and decimals?

A: Yes, you can use scientific notation to represent numbers with negative exponents and decimals. For example, the number $2.75 \times 10^{-6}$ can be written in scientific notation as $2.75 \times 10^{-6}$.

Q: How do I check my work when dividing numbers in scientific notation with exponents and decimals?

A: To check your work when dividing numbers in scientific notation with exponents and decimals, you need to make sure that the result is expressed in the correct units and that the calculation is correct.

Q: Can I use scientific notation to represent numbers with fractions and decimals?

A: Yes, you can use scientific notation to represent numbers with fractions and decimals. For example, the number $2.75 \times 10^{-6}$ can be written in scientific notation as $2.75 \times 10^{-6}$.

Q: How do I simplify a number in scientific notation with fractions and decimals?

A: To simplify a number in scientific notation with fractions and decimals, you need to multiply the number by the reciprocal of the fraction and then simplify the result.

Q: Can I use scientific notation to represent numbers with exponents, decimals, and fractions?

A: Yes, you can use scientific notation to represent numbers with exponents, decimals, and fractions. For example, the number $2.75 \times 10^{-6}$ can be written in scientific notation as $2.75 \times 10^{-6}$.

Q: How do I check my units when dividing numbers in scientific notation with exponents, decimals, and fractions?

A: To check your units when dividing numbers in scientific notation with exponents, decimals, and fractions, you need to make sure that the units are consistent and that the result is expressed in the correct units.

Q: Can I use scientific notation to represent numbers with negative exponents, decimals, and fractions?

A: Yes, you can use scientific notation to represent numbers with negative exponents, decimals, and fractions. For example, the number $2.75 \times 10^{-6}$ can be written