Select The Correct Answer.Which Of The Following Numbers Is Equivalent To $5.1 \times 10^{-2}$?A. 0.0051 B. 0.051 C. 510 D. 5,100

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. In this article, we will explore the concept of scientific notation and how to convert numbers from scientific notation to standard decimal form.

What is Scientific Notation?

Scientific notation is a method of expressing numbers as a product of a number between 1 and 10 and a power of 10. The general form of a number in scientific notation is:

$$a \times 10^n$$

where $a$ is a number between 1 and 10, and $n$ is an integer that represents the power of 10.

Example of Scientific Notation

Let's consider the number 500,000. We can express this number in scientific notation as:

$$5 \times 10^5$$

In this example, the number 5 is between 1 and 10, and the power of 10 is 5.

Converting Scientific Notation to Standard Decimal Form

To convert a number from scientific notation to standard decimal form, we need to multiply the number between 1 and 10 by the corresponding power of 10.

Step 1: Multiply the Number

The first step is to multiply the number between 1 and 10 by the corresponding power of 10.

Step 2: Add the Exponents

If the power of 10 is positive, we add the exponents. If the power of 10 is negative, we subtract the exponents.

Step 3: Write the Result

The final step is to write the result as a standard decimal number.

Example: Converting 5.1 × 10^−2 to Standard Decimal Form

Let's consider the number 5.1 × 10^−2. To convert this number to standard decimal form, we need to multiply 5.1 by 10^−2.

Step 1: Multiply the Number

$$5.1 \times 10^{-2} = 0.051$$

Step 2: Add the Exponents

In this case, we don't need to add the exponents because the power of 10 is negative.

Step 3: Write the Result

The final result is 0.051.

Conclusion

In conclusion, scientific notation is a way of expressing very large or very small numbers in a more manageable form. To convert a number from scientific notation to standard decimal form, we need to multiply the number between 1 and 10 by the corresponding power of 10. We can then add the exponents if the power of 10 is positive, or subtract the exponents if the power of 10 is negative. Finally, we can write the result as a standard decimal number.

Answer

The correct answer is B. 0.051.

Discussion

Scientific notation is a useful tool for expressing very large or very small numbers. It is commonly used in mathematics, physics, and engineering. However, it can be challenging to convert numbers from scientific notation to standard decimal form. By following the steps outlined in this article, we can easily convert numbers from scientific notation to standard decimal form.

Common Mistakes

When converting numbers from scientific notation to standard decimal form, it is easy to make mistakes. Here are some common mistakes to avoid:

• Incorrect Multiplication: Make sure to multiply the number between 1 and 10 by the corresponding power of 10.
• Incorrect Addition or Subtraction of Exponents: Make sure to add the exponents if the power of 10 is positive, or subtract the exponents if the power of 10 is negative.
• Incorrect Writing of the Result: Make sure to write the result as a standard decimal number.

Practice Problems

Here are some practice problems to help you master the conversion of numbers from scientific notation to standard decimal form:

1. Convert 3.2 × 10^3 to standard decimal form.
2. Convert 9.5 × 10^−1 to standard decimal form.
3. Convert 2.1 × 10^2 to standard decimal form.

Answer Key

1. 3200
2. 0.95
3. 210

Conclusion

Frequently Asked Questions

Scientific notation is a way of expressing very large or very small numbers in a more manageable form. Here are some frequently asked questions about scientific notation:

Q: What is scientific notation?

A: Scientific notation is a way of expressing numbers as a product of a number between 1 and 10 and a power of 10.

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

A: To convert a number from scientific notation to standard decimal form, you need to multiply the number between 1 and 10 by the corresponding power of 10. If the power of 10 is positive, you add the exponents. If the power of 10 is negative, you subtract the exponents.

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

A: Scientific notation is a way of expressing numbers as a product of a number between 1 and 10 and a power of 10. Standard decimal form is a way of expressing numbers as a decimal number with a whole number part and a fractional part.

Q: When should I use scientific notation?

A: You should use scientific notation when you need to express very large or very small numbers in a more manageable form.

Q: How do I know when to use positive or negative exponents?

A: If the number is greater than 10, you use a positive exponent. If the number is less than 1, you use a negative exponent.

Q: Can I use scientific notation with fractions?

A: Yes, you can use scientific notation with fractions. For example, 0.0005 can be written as 5 × 10^−4.

Q: Can I use scientific notation with decimals?

A: Yes, you can use scientific notation with decimals. For example, 0.05 can be written as 5 × 10^−2.

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

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

Q: What are some common mistakes to avoid when using scientific notation?

A: Some common mistakes to avoid when using scientific notation include:

• Incorrect multiplication: Make sure to multiply the number between 1 and 10 by the corresponding power of 10.
• Incorrect addition or subtraction of exponents: Make sure to add the exponents if the power of 10 is positive, or subtract the exponents if the power of 10 is negative.
• Incorrect writing of the result: Make sure to write the result as a standard decimal number.

Q: How do I use scientific notation in real-life situations?

A: Scientific notation is commonly used in mathematics, physics, and engineering. It is also used in everyday life when dealing with very large or very small numbers, such as in finance, science, and technology.

Conclusion

In conclusion, scientific notation is a useful tool for expressing very large or very small numbers in a more manageable form. By following the steps outlined in this article, you can easily convert numbers from scientific notation to standard decimal form and vice versa. Remember to multiply the number between 1 and 10 by the corresponding power of 10, add the exponents if the power of 10 is positive, or subtract the exponents if the power of 10 is negative, and write the result as a standard decimal number.