Which Expression Is Equivalent To $5.42 \times 10^{-4}$?A. 0.000542 B. 5,420,000 C. 0.0000542 D. 54,200

Feb 28, 2025 by ADMIN 111 views

**Understanding Scientific Notation and Equivalent Expressions**

Introduction to Scientific Notation

Scientific notation is a way of expressing very large or very small numbers in a more manageable form. It consists of a number between 1 and 10, multiplied by a power of 10. This notation is commonly used in mathematics, physics, and engineering to simplify calculations and make it easier to understand and compare large or small numbers.

Understanding the Given Expression

The given expression is $5.42 \times 10^{-4}$. This means that the number 5.42 is multiplied by 10 to the power of -4. To understand this expression, we need to know that the exponent -4 indicates that the number is very small, and the decimal point is shifted 4 places to the left.

Analyzing the Options

Now, let's analyze the options given to determine which one is equivalent to the given expression.

Option A: 0.000542

This option is a decimal number with 6 digits. To compare it with the given expression, we need to convert it to scientific notation. We can rewrite 0.000542 as $5.42 \times 10^{-4}$. This means that option A is equivalent to the given expression.

Option B: 5,420,000

This option is a large number with 7 digits. To compare it with the given expression, we need to convert it to scientific notation. We can rewrite 5,420,000 as $5.42 \times 10^{6}$. This means that option B is not equivalent to the given expression.

Option C: 0.0000542

This option is a decimal number with 7 digits. To compare it with the given expression, we need to convert it to scientific notation. We can rewrite 0.0000542 as $5.42 \times 10^{-5}$. This means that option C is not equivalent to the given expression.

Option D: 54,200

This option is a large number with 5 digits. To compare it with the given expression, we need to convert it to scientific notation. We can rewrite 54,200 as $5.42 \times 10^{4}$. This means that option D is not equivalent to the given expression.

Conclusion

Based on the analysis of the options, we can conclude that option A, 0.000542, is the only one that is equivalent to the given expression $5.42 \times 10^{-4}$. This means that the correct answer is option A.

Understanding the Importance of Scientific Notation

Scientific notation is an essential concept in mathematics and science. It allows us to express very large or very small numbers in a more manageable form, making it easier to understand and compare them. By understanding scientific notation, we can perform calculations and solve problems more efficiently.

Real-World Applications of Scientific Notation

Scientific notation has many real-world applications. It is used in various fields such as physics, engineering, and chemistry to express quantities such as distances, speeds, and temperatures. For example, the distance between the Earth and the Sun is approximately $1.5 \times 10^{11}$. This means that the distance is 1.5 followed by 11 zeros.

Tips for Working with Scientific Notation

When working with scientific notation, it's essential to follow some tips to avoid errors. Here are some tips:

Use the correct exponent: Make sure to use the correct exponent when converting a number to scientific notation.
Use the correct decimal point: Make sure to use the correct decimal point when converting a number to scientific notation.
Use the correct multiplication: Make sure to use the correct multiplication when multiplying numbers in scientific notation.
Use the correct division: Make sure to use the correct division when dividing numbers in scientific notation.

Conclusion

In conclusion, scientific notation is an essential concept in mathematics and science. It allows us to express very large or very small numbers in a more manageable form, making it easier to understand and compare them. By understanding scientific notation, we can perform calculations and solve problems more efficiently.

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 consists of a number between 1 and 10, multiplied by a power of 10.

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, you need to multiply the number by 10 raised to the power of the number of places you moved the decimal point.

Q: What is the exponent in scientific notation?

A: The exponent in scientific notation is the power of 10 that the number is multiplied by. It indicates whether the number is very large or very small.

Q: How do I multiply numbers in scientific notation?

A: To multiply numbers in scientific notation, you need to multiply the numbers and add the exponents. For example, $3.4 \times 10^2 \times 2.5 \times 10^3 = 8.5 \times 10^5$.

Q: How do I divide numbers in scientific notation?

A: To divide numbers in scientific notation, you need to divide the numbers and subtract the exponents. For example, $3.4 \times 10^2 \div 2.5 \times 10^3 = 1.36 \times 10^{-1}$.

Q: What is the difference between scientific notation and standard notation?

A: Scientific notation and standard notation are two different ways of expressing numbers. Scientific notation is used to express very large or very small numbers, while standard notation is used to express numbers in their standard form.

Q: When should I use scientific notation?

A: You should use scientific notation when you need to express very large or very small numbers. It makes it easier to understand and compare these numbers.

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. For example, $3.4 \times 10^2 = 340$.

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

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

Q: What are some common applications of scientific notation?

A: Scientific notation is used in various fields such as physics, engineering, and chemistry to express quantities such as distances, speeds, and temperatures.

Q: How do I use scientific notation in real-world problems?

A: You can use scientific notation in real-world problems by expressing quantities such as distances, speeds, and temperatures in a more manageable form. For example, the distance between the Earth and the Sun is approximately $1.5 \times 10^{11}$.

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:

Using the wrong exponent: Make sure to use the correct exponent when converting a number to scientific notation.
Using the wrong decimal point: Make sure to use the correct decimal point when converting a number to scientific notation.
Using the wrong multiplication: Make sure to use the correct multiplication when multiplying numbers in scientific notation.
Using the wrong division: Make sure to use the correct division when dividing numbers in scientific notation.