What Is $2.3 \times 10^{14}$ Divided By $5.4 \times 10^{-3}$?A. \$4.3 \times 10^{10}$[/tex\] B. $1.4 \times 10^{11}$ C. $4.3 \times 10^{16}$ D. \$1.2 \times 10^{12}$[/tex\]
What is $2.3 \times 10^{14}$ Divided by $5.4 \times 10^{-3}$?
Understanding Exponents and Division
When dealing with numbers in scientific notation, it's essential to understand the rules of exponents and how to perform operations like division. In this case, we're asked to find the result of dividing $2.3 \times 10^{14}$ by $5.4 \times 10^{-3}$.
The Rules of Exponents
To divide numbers in scientific notation, we need to follow the rules of exponents. When dividing two numbers in scientific notation, we divide the coefficients (the numbers in front of the exponents) and subtract the exponents. This is based on the rule that $a^m \div a^n = a^{m-n}$.
Applying the Rules of Exponents
Let's apply the rules of exponents to the given problem. We have:
To divide the coefficients, we simply divide 2.3 by 5.4, which gives us:
However, we're not done yet. We also need to subtract the exponents. The exponent of the dividend is 14, and the exponent of the divisor is -3. Subtracting the exponents, we get:
So, the result of dividing the coefficients is 0.4263, and the result of subtracting the exponents is 17. Combining these two results, we get:
Simplifying the Result
Now that we have the result of the division, we can simplify it by expressing it in scientific notation. To do this, we need to move the decimal point of the coefficient to the left until we have a number between 1 and 10. In this case, we need to move the decimal point 2 places to the left, which gives us:
Comparing the Result to the Answer Choices
Now that we have the result of the division, we can compare it to the answer choices. The answer choices are:
A. $4.3 \times 10^{10}$ B. $1.4 \times 10^{11}$ C. $4.3 \times 10^{16}$ D. $1.2 \times 10^{12}$
Comparing our result to the answer choices, we see that the correct answer is:
C. $4.3 \times 10^{16}$
Conclusion
In this article, we learned how to divide numbers in scientific notation by applying the rules of exponents. We saw that when dividing two numbers in scientific notation, we divide the coefficients and subtract the exponents. We then applied this rule to the given problem and simplified the result to express it in scientific notation. Finally, we compared our result to the answer choices and found that the correct answer is C. $4.3 \times 10^{16}$.
Understanding the Importance of Scientific Notation
Scientific notation is a powerful tool for expressing very large or very small numbers in a concise and manageable way. By understanding the rules of exponents and how to perform operations like division, we can solve complex problems and make sense of the world around us.
Real-World Applications of Scientific Notation
Scientific notation has many real-world applications, from physics and engineering to finance and economics. For example, scientists use scientific notation to express the size of atoms and molecules, while engineers use it to calculate the stresses and strains on buildings and bridges. In finance, scientific notation is used to express the value of large sums of money, and in economics, it's used to calculate the growth rates of economies.
Conclusion
In conclusion, scientific notation is a powerful tool for expressing very large or very small numbers in a concise and manageable way. By understanding the rules of exponents and how to perform operations like division, we can solve complex problems and make sense of the world around us. Whether we're scientists, engineers, or simply curious individuals, scientific notation is an essential tool for navigating the complexities of the world.
Final Answer
The final answer is C. $4.3 \times 10^{16}$.
Q&A: Scientific Notation and Exponents
Understanding Scientific Notation and Exponents
Scientific notation and exponents are fundamental concepts in mathematics that help us express very large or very small numbers in a concise and manageable way. In this article, we'll answer some frequently asked questions about scientific notation and exponents to help you better understand these concepts.
Q: What is scientific notation?
A: Scientific notation is a way of expressing very large or very small numbers in a concise and manageable way. 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 move the decimal point of the number to the left until you have a number between 1 and 10. Then, you multiply the number by a power of 10 that is equal to the number of places you moved the decimal point.
Q: What is the rule for multiplying numbers in scientific notation?
A: When multiplying numbers in scientific notation, you multiply the coefficients (the numbers in front of the exponents) and add the exponents.
Q: What is the rule for dividing numbers in scientific notation?
A: When dividing numbers in scientific notation, you divide the coefficients (the numbers in front of the exponents) and subtract the exponents.
Q: How do I simplify a number in scientific notation?
A: To simplify a number in scientific notation, you need to move the decimal point of the coefficient to the left until you have a number between 1 and 10. Then, you multiply the coefficient by a power of 10 that is equal to the number of places you moved the decimal point.
Q: What is the difference between a positive exponent and a negative exponent?
A: A positive exponent indicates that the number is being multiplied by itself a certain number of times, while a negative exponent indicates that the number is being divided by itself a certain number of times.
Q: How do I evaluate an expression with exponents?
A: To evaluate an expression with exponents, you need to follow the order of operations (PEMDAS):
- Evaluate any expressions inside parentheses.
- Evaluate any exponential expressions (e.g. 2^3).
- Evaluate any multiplication and division expressions from left to right.
- Evaluate any addition and subtraction expressions from left to right.
Q: What is the rule for raising a power to a power?
A: When raising a power to a power, you multiply the exponents.
Q: What is the rule for multiplying a power by a number?
A: When multiplying a power by a number, you multiply the exponent by the number.
Q: What is the rule for dividing a power by a number?
A: When dividing a power by a number, you divide the exponent by the number.
Q: How do I simplify an expression with exponents?
A: To simplify an expression with exponents, you need to follow the rules for exponents and simplify the expression as much as possible.
Conclusion
In this article, we've answered some frequently asked questions about scientific notation and exponents to help you better understand these concepts. By following the rules for scientific notation and exponents, you can simplify complex expressions and make sense of the world around you.
Final Tips
- Always follow the order of operations (PEMDAS) when evaluating expressions with exponents.
- Use the rules for exponents to simplify expressions and make them easier to evaluate.
- Practice, practice, practice! The more you practice working with scientific notation and exponents, the more comfortable you'll become with these concepts.
Additional Resources
- Khan Academy: Scientific Notation and Exponents
- Mathway: Scientific Notation and Exponents
- Wolfram Alpha: Scientific Notation and Exponents
Final Answer
The final answer is C. $4.3 \times 10^{16}$.