Which Number Is Greatest?A. 8.23 × 10 12 8.23 \times 10^{12} 8.23 × 1 0 12 B. 8.28 × 10 8 8.28 \times 10^8 8.28 × 1 0 8 C. 8.23 × 10 − 8 8.23 \times 10^{-8} 8.23 × 1 0 − 8 D. 823 × 10 3 823 \times 10^3 823 × 1 0 3

Introduction

In mathematics, comparing numbers with different exponents can be a challenging task. When numbers are expressed in scientific notation, it's essential to understand the concept of exponents and how they affect the magnitude of a number. In this article, we will delve into the world of scientific notation and explore which number is greatest among the given options.

Understanding Scientific Notation

Scientific notation is a way of expressing numbers in the form of a product of a number between 1 and 10 and a power of 10. This notation is commonly used in mathematics, physics, and engineering to represent very large or very small numbers in a more manageable and concise form.

For example, the number 456,000 can be expressed in scientific notation as 4.56 × 10^5. Similarly, the number 0.000456 can be expressed as 4.56 × 10^-4.

Analyzing the Options

Now that we have a basic understanding of scientific notation, let's analyze the given options:

A. $8.23 \times 10^{12}$ B. $8.28 \times 10^8$ C. $8.23 \times 10^{-8}$ D. $823 \times 10^3$

Option A: $8.23 \times 10^{12}$

The exponent in this option is 12, which is a very large number. To understand the magnitude of this number, let's consider that 10^12 is equal to 1,000,000,000,000. Therefore, $8.23 \times 10^{12}$ is equivalent to 8,230,000,000,000.

Option B: $8.28 \times 10^8$

The exponent in this option is 8, which is significantly smaller than 12. To understand the magnitude of this number, let's consider that 10^8 is equal to 100,000,000. Therefore, $8.28 \times 10^8$ is equivalent to 828,000,000.

Option C: $8.23 \times 10^{-8}$

The exponent in this option is -8, which is a very small number. To understand the magnitude of this number, let's consider that 10^-8 is equal to 0.00000001. Therefore, $8.23 \times 10^{-8}$ is equivalent to 0.0000000823.

Option D: $823 \times 10^3$

The exponent in this option is 3, which is significantly smaller than 12. To understand the magnitude of this number, let's consider that 10^3 is equal to 1,000. Therefore, $823 \times 10^3$ is equivalent to 823,000.

Comparing the Numbers

Now that we have analyzed each option, let's compare the numbers:

• $8.23 \times 10^{12}$ is equivalent to 8,230,000,000,000
• $8.28 \times 10^8$ is equivalent to 828,000,000
• $8.23 \times 10^{-8}$ is equivalent to 0.0000000823
• $823 \times 10^3$ is equivalent to 823,000

Based on the analysis, it's clear that option A, $8.23 \times 10^{12}$, is the greatest number among the given options.

Conclusion

In conclusion, comparing numbers with different exponents requires a thorough understanding of scientific notation and the concept of exponents. By analyzing each option and understanding the magnitude of each number, we can determine which number is greatest. In this case, option A, $8.23 \times 10^{12}$, is the greatest number among the given options.

Key Takeaways

• Scientific notation is a way of expressing numbers in the form of a product of a number between 1 and 10 and a power of 10.
• Exponents affect the magnitude of a number, with larger exponents resulting in larger numbers.
• Comparing numbers with different exponents requires a thorough understanding of scientific notation and the concept of exponents.

Frequently Asked Questions

Q: What is scientific notation? A: Scientific notation is a way of expressing numbers in the form of a product of a number between 1 and 10 and a power of 10.

Q: How do exponents affect the magnitude of a number? A: Exponents affect the magnitude of a number, with larger exponents resulting in larger numbers.

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Q: What is scientific notation?

A: Scientific notation is a way of expressing numbers in the form of a product of a number between 1 and 10 and a power of 10. It is commonly used in mathematics, physics, and engineering to represent very large or very small numbers in a more manageable and concise form.

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

A: To convert a number to scientific notation, follow these steps:

1. Move the decimal point to the left until you have a number between 1 and 10.
2. Count the number of places you moved the decimal point.
3. Write the number in the form of a product of a number between 1 and 10 and a power of 10.

For example, to convert the number 456,000 to scientific notation, move the decimal point 5 places to the left to get 4.56. Then, count the number of places you moved the decimal point and write the number in the form of a product of a number between 1 and 10 and a power of 10: 4.56 × 10^5.

Q: What is the difference between a positive and negative exponent?

A: A positive exponent indicates that the number is being multiplied by 10 raised to that power. A negative exponent indicates that the number is being divided by 10 raised to that power.

For example, 4.56 × 10^5 is equivalent to 456,000, while 4.56 × 10^-5 is equivalent to 0.0000456.

Q: How do I compare numbers with different exponents?

A: To compare numbers with different exponents, compare the exponents first. If the exponents are the same, compare the numbers themselves. If the exponents are different, the number with the larger exponent is greater.

For example, to compare the numbers 4.56 × 10^5 and 4.56 × 10^3, compare the exponents first. Since 5 is greater than 3, the number 4.56 × 10^5 is greater.

Q: What is the order of operations for exponents?

A: The order of operations for exponents is:

1. Evaluate any expressions inside parentheses.
2. Evaluate any exponents (e.g. 2^3).
3. Evaluate any multiplication and division operations from left to right.
4. Evaluate any addition and subtraction operations from left to right.

For example, to evaluate the expression 2^3 × 4 + 5, follow the order of operations:

1. Evaluate the exponent: 2^3 = 8.
2. Multiply 8 by 4: 8 × 4 = 32.
3. Add 5 to 32: 32 + 5 = 37.

Q: Can I simplify an expression with exponents?

A: Yes, you can simplify an expression with exponents by combining like terms.

For example, to simplify the expression 2^3 × 2^2, combine the like terms:

2^3 × 2^2 = 2^(3+2) = 2^5

Q: What are some common mistakes to avoid when working with exponents?

A: Some common mistakes to avoid when working with exponents include:

• Forgetting to evaluate exponents before multiplying or dividing.
• Forgetting to combine like terms.
• Making errors when evaluating expressions with multiple exponents.

Q: How can I practice working with exponents?

A: You can practice working with exponents by:

• Solving problems and exercises in a textbook or online resource.
• Creating your own problems and exercises to practice.
• Working with a tutor or teacher to get help and feedback.

Conclusion

In conclusion, working with exponents can be a challenging but rewarding topic. By understanding the basics of exponents and practicing with problems and exercises, you can become more confident and proficient in your ability to work with exponents. Remember to always follow the order of operations and to combine like terms to simplify expressions. With practice and patience, you can master the art of working with exponents.