Evaluate The Following Expression:\[$(0.0034)\left(3.1 \times 10^5\right)\$\](a) \[$1.054 \times 10^9\$\](b) 0.01054(c) 1054(d) \[$1054 \times 10^5\$\]
Understanding Scientific Notation
Scientific notation is a way of expressing very large or very small numbers in a compact form. It consists of a number between 1 and 10 multiplied by a power of 10. For example, the number 100 can be written in scientific notation as 1 × 10^2, while the number 0.01 can be written as 1 × 10^-2.
Evaluating the Given Expression
The given expression is: {(0.0034)\left(3.1 \times 10^5\right)$}$
To evaluate this expression, we need to follow the order of operations (PEMDAS):
- Multiply 0.0034 by 3.1
- Multiply the result by 10^5
Step 1: Multiply 0.0034 by 3.1
First, we multiply 0.0034 by 3.1:
0.0034 × 3.1 = 0.01054
Step 2: Multiply the Result by 10^5
Next, we multiply the result by 10^5:
0.01054 × 10^5 = 1.054 × 10^6
Evaluating the Options
Now, let's evaluate the options:
(a) ${1.054 \times 10^9\$}
This option is incorrect because the correct result is 1.054 × 10^6, not 1.054 × 10^9.
(b) 0.01054
This option is incorrect because the correct result is 1.054 × 10^6, not 0.01054.
(c) 1054
This option is incorrect because the correct result is 1.054 × 10^6, not 1054.
(d) ${1054 \times 10^5\$}
This option is incorrect because the correct result is 1.054 × 10^6, not 1054 × 10^5.
Conclusion
The correct answer is not among the options provided. However, we can rewrite the correct result in a more compact form:
1.054 × 10^6 = 1.054 × 10^6
This is the correct result of the given expression.
Tips and Tricks
When working with scientific notation, it's essential to follow the order of operations (PEMDAS) and to be careful with the powers of 10.
- When multiplying numbers in scientific notation, multiply the coefficients (the numbers in front of the powers of 10) and add the exponents.
- When dividing numbers in scientific notation, divide the coefficients and subtract the exponents.
By following these tips and tricks, you can become more confident and proficient in evaluating scientific notation expressions.
Common Mistakes to Avoid
When working with scientific notation, it's easy to make mistakes. Here are some common mistakes to avoid:
- Forgetting to follow the order of operations (PEMDAS)
- Not being careful with the powers of 10
- Not multiplying or dividing the coefficients correctly
- Not adding or subtracting the exponents correctly
By being aware of these common mistakes, you can avoid them and become more accurate in your calculations.
Real-World Applications
Scientific notation has many real-world applications, including:
- Calculating large or small numbers in physics, chemistry, and engineering
- Expressing very large or very small numbers in computer science and data analysis
- Performing calculations with very large or very small numbers in finance and economics
By understanding scientific notation and how to evaluate expressions in this notation, you can become more proficient in a wide range of fields and applications.
Conclusion
Q: What is scientific notation?
A: Scientific notation is a way of expressing very large or very small numbers in a compact form. It consists of a number between 1 and 10 multiplied by a power of 10.
Q: How do I write a number in scientific notation?
A: To write a number in scientific notation, you need to express it as a number between 1 and 10 multiplied by a power of 10. For example, the number 100 can be written in scientific notation as 1 × 10^2, while the number 0.01 can be written as 1 × 10^-2.
Q: How do I evaluate an expression in scientific notation?
A: To evaluate an expression in scientific notation, you need to follow the order of operations (PEMDAS):
- Multiply or divide the coefficients (the numbers in front of the powers of 10)
- Add or subtract the exponents (the powers of 10)
Q: What is the order of operations (PEMDAS)?
A: The order of operations (PEMDAS) is a set of rules that tells you which operations to perform first when evaluating an expression. The acronym PEMDAS stands for:
- Parentheses: Evaluate expressions inside parentheses first
- Exponents: Evaluate any exponential expressions next (e.g. 2^3)
- Multiplication and Division: Evaluate any multiplication and division operations from left to right
- Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right
Q: How do I multiply numbers in scientific notation?
A: To multiply numbers in scientific notation, you need to multiply the coefficients (the numbers in front of the powers of 10) and add the exponents (the powers of 10). For example:
(2 × 10^3) × (3 × 10^4) = (2 × 3) × (10^3 × 10^4) = 6 × 10^7
Q: How do I divide numbers in scientific notation?
A: To divide numbers in scientific notation, you need to divide the coefficients (the numbers in front of the powers of 10) and subtract the exponents (the powers of 10). For example:
(2 × 10^3) ÷ (3 × 10^4) = (2 ÷ 3) × (10^3 ÷ 10^4) = 0.67 × 10^(-1)
Q: What are some common mistakes to avoid when working with scientific notation?
A: Some common mistakes to avoid when working with scientific notation include:
- Forgetting to follow the order of operations (PEMDAS)
- Not being careful with the powers of 10
- Not multiplying or dividing the coefficients correctly
- Not adding or subtracting the exponents correctly
Q: What are some real-world applications of scientific notation?
A: Scientific notation has many real-world applications, including:
- Calculating large or small numbers in physics, chemistry, and engineering
- Expressing very large or very small numbers in computer science and data analysis
- Performing calculations with very large or very small numbers in finance and economics
Q: How can I practice evaluating scientific notation expressions?
A: You can practice evaluating scientific notation expressions by working through examples and exercises. You can also try using online resources or calculators to help you evaluate expressions.
Conclusion
Evaluating scientific notation expressions requires careful attention to the order of operations (PEMDAS) and the powers of 10. By following these tips and tricks, you can become more confident and proficient in evaluating scientific notation expressions. Remember to avoid common mistakes and to be aware of the real-world applications of scientific notation. With practice and patience, you can master the art of evaluating scientific notation expressions.