Simplify The Following Expression. Write Your Answer In Scientific Notation.$\left(4.2 \cdot 10^6\right) - \left(6.9 \cdot 10^6\right$\]
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 complex calculations. In this article, we will explore how to simplify expressions in scientific notation, with a focus on subtraction.
Understanding Scientific Notation
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 × 10^5. This notation is useful because it allows us to easily compare and manipulate numbers that are very large or very small.
Simplifying Expressions in Scientific Notation
When simplifying expressions in scientific notation, we need to follow the rules of arithmetic operations. In this case, we are dealing with subtraction. To simplify the expression (4.2 × 10^6) - (6.9 × 10^6), we need to subtract the two numbers.
Step 1: Subtract the Coefficients
The first step is to subtract the coefficients of the two numbers. In this case, we have 4.2 and 6.9. To subtract these numbers, we need to make sure that they have the same exponent. Since they both have an exponent of 10^6, we can proceed with the subtraction.
4.2 - 6.9 = -2.7
Step 2: Subtract the Exponents
Now that we have subtracted the coefficients, we need to subtract the exponents. In this case, we have 10^6 and 10^6. To subtract these exponents, we need to subtract the powers of 10.
10^6 - 10^6 = 0
Step 3: Combine the Results
Now that we have subtracted the coefficients and the exponents, we can combine the results. In this case, we have -2.7 and 0. Since the exponent is 0, we can ignore it and simply write the result as -2.7.
Conclusion
In conclusion, simplifying expressions in scientific notation involves following the rules of arithmetic operations. When dealing with subtraction, we need to subtract the coefficients and the exponents separately. In this case, we simplified the expression (4.2 × 10^6) - (6.9 × 10^6) to -2.7 × 10^6.
Example 1: Simplifying an Expression with a Negative Exponent
Let's consider another example. Suppose we have the expression (3.4 × 10^-3) - (2.1 × 10^-3). To simplify this expression, we need to follow the same steps as before.
Step 1: Subtract the Coefficients
The first step is to subtract the coefficients of the two numbers. In this case, we have 3.4 and 2.1.
3.4 - 2.1 = 1.3
Step 2: Subtract the Exponents
Now that we have subtracted the coefficients, we need to subtract the exponents. In this case, we have 10^-3 and 10^-3. To subtract these exponents, we need to subtract the powers of 10.
10^-3 - 10^-3 = 0
Step 3: Combine the Results
Now that we have subtracted the coefficients and the exponents, we can combine the results. In this case, we have 1.3 and 0. Since the exponent is 0, we can ignore it and simply write the result as 1.3 × 10^-3.
Example 2: Simplifying an Expression with a Large Exponent
Let's consider another example. Suppose we have the expression (9.8 × 10^8) - (4.5 × 10^8). To simplify this expression, we need to follow the same steps as before.
Step 1: Subtract the Coefficients
The first step is to subtract the coefficients of the two numbers. In this case, we have 9.8 and 4.5.
9.8 - 4.5 = 5.3
Step 2: Subtract the Exponents
Now that we have subtracted the coefficients, we need to subtract the exponents. In this case, we have 10^8 and 10^8. To subtract these exponents, we need to subtract the powers of 10.
10^8 - 10^8 = 0
Step 3: Combine the Results
Now that we have subtracted the coefficients and the exponents, we can combine the results. In this case, we have 5.3 and 0. Since the exponent is 0, we can ignore it and simply write the result as 5.3 × 10^8.
Conclusion
In conclusion, simplifying expressions in scientific notation involves following the rules of arithmetic operations. When dealing with subtraction, we need to subtract the coefficients and the exponents separately. By following these steps, we can simplify complex expressions and arrive at the correct result.
Tips and Tricks
Here are some tips and tricks to help you simplify expressions in scientific notation:
- Make sure to follow the rules of arithmetic operations when simplifying expressions in scientific notation.
- When dealing with subtraction, subtract the coefficients and the exponents separately.
- Use the correct notation when expressing numbers in scientific notation.
- Practice simplifying expressions in scientific notation to become more comfortable with the process.
Common Mistakes
Here are some common mistakes to avoid when simplifying expressions in scientific notation:
- Failing to follow the rules of arithmetic operations.
- Subtracting the exponents before subtracting the coefficients.
- Using the wrong notation when expressing numbers in scientific notation.
- Not practicing simplifying expressions in scientific notation.
Conclusion
Introduction
In our previous article, we explored how to simplify expressions in scientific notation. We discussed the rules of arithmetic operations and provided examples of how to simplify expressions with different exponents. In this article, we will answer some frequently asked questions about simplifying expressions 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 complex calculations.
Q: How do I simplify an expression in scientific notation?
A: To simplify an expression in scientific notation, you need to follow the rules of arithmetic operations. When dealing with subtraction, subtract the coefficients and the exponents separately.
Q: What is the difference between a coefficient and an exponent?
A: A coefficient is a number that is multiplied by a power of 10, while an exponent is the power to which the 10 is raised.
Q: How do I subtract exponents?
A: To subtract exponents, you need to subtract the powers of 10. For example, 10^6 - 10^6 = 0.
Q: Can I simplify an expression with a negative exponent?
A: Yes, you can simplify an expression with a negative exponent. To do this, you need to follow the same steps as before, but be careful when subtracting the exponents.
Q: What is the correct notation for expressing numbers in scientific notation?
A: The correct notation for expressing numbers in scientific notation is a number between 1 and 10, multiplied by a power of 10. For example, 456,000 can be expressed in scientific notation as 4.56 × 10^5.
Q: How do I simplify an expression with a large exponent?
A: To simplify an expression with a large exponent, you need to follow the same steps as before. Be careful when subtracting the exponents, as the result may be a very large or very small number.
Q: Can I simplify an expression with a decimal exponent?
A: Yes, you can simplify an expression with a decimal exponent. To do this, you need to follow the same steps as before, but be careful when subtracting the exponents.
Q: What are some common mistakes to avoid when simplifying expressions in scientific notation?
A: Some common mistakes to avoid when simplifying expressions in scientific notation include:
- Failing to follow the rules of arithmetic operations
- Subtracting the exponents before subtracting the coefficients
- Using the wrong notation when expressing numbers in scientific notation
- Not practicing simplifying expressions in scientific notation
Q: How can I practice simplifying expressions in scientific notation?
A: You can practice simplifying expressions in scientific notation by working through examples and exercises. You can also try simplifying expressions with different exponents and coefficients to become more comfortable with the process.
Conclusion
In conclusion, simplifying expressions in scientific notation is an important skill to have in mathematics and science. By following the rules of arithmetic operations and practicing simplifying expressions, you can become more comfortable with the process and arrive at the correct result. Remember to avoid common mistakes and use the correct notation when expressing numbers in scientific notation.
Additional Resources
If you are looking for additional resources to help you learn about simplifying expressions in scientific notation, here are a few suggestions:
- Khan Academy: Scientific Notation
- Mathway: Scientific Notation
- Wolfram Alpha: Scientific Notation
Conclusion
In conclusion, simplifying expressions in scientific notation is a valuable skill that can be applied to a wide range of mathematical and scientific problems. By following the rules of arithmetic operations and practicing simplifying expressions, you can become more comfortable with the process and arrive at the correct result. Remember to avoid common mistakes and use the correct notation when expressing numbers in scientific notation.