{ \left(6.8 \times 10^7\right) - \left(2.3 \times 10^7\right)$}$Write Your Answer In Scientific Notation. { \square$}$

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Scientific notation is a way of expressing very large or very small numbers in a more manageable form. It is a fundamental concept in mathematics and is widely used in various fields, including physics, engineering, and chemistry. In this article, we will discuss the basics of scientific notation and how to calculate expressions involving large numbers.

What is Scientific Notation?

Scientific notation is a way of expressing a number as a product of a number between 1 and 10 and a power of 10. It is written in the form:

a × 10^n

where a is the coefficient and n is the exponent. For example, the number 456,000 can be written in scientific notation as 4.56 × 10^5.

Understanding Exponents

Exponents are a crucial part of scientific notation. They represent the power to which a number is raised. In the example above, the exponent is 5, which means that 10 is raised to the power of 5. Exponents can be positive or negative, and they can be used to represent very large or very small numbers.

Calculating Expressions in Scientific Notation

Now that we have a basic understanding of scientific notation and exponents, let's move on to calculating expressions involving large numbers. The expression given in the problem is:

(6.8×107)−(2.3×107)\left(6.8 \times 10^7\right) - \left(2.3 \times 10^7\right)

To calculate this expression, we need to follow the order of operations (PEMDAS):

  1. Parentheses: Evaluate the expressions inside the parentheses.
  2. Exponents: Evaluate the exponents.
  3. Multiplication and Division: Evaluate the multiplication and division operations from left to right.
  4. Addition and Subtraction: Evaluate the addition and subtraction operations from left to right.

Step 1: Evaluate the Expressions Inside the Parentheses

The expressions inside the parentheses are:

(6.8×107)\left(6.8 \times 10^7\right)

and

(2.3×107)\left(2.3 \times 10^7\right)

These expressions are already in scientific notation, so we don't need to do anything else.

Step 2: Subtract the Two Expressions

Now that we have evaluated the expressions inside the parentheses, we can subtract the two expressions:

(6.8×107)−(2.3×107)\left(6.8 \times 10^7\right) - \left(2.3 \times 10^7\right)

To subtract these expressions, we need to subtract the coefficients and the exponents separately.

Step 3: Subtract the Coefficients

The coefficients are 6.8 and 2.3. To subtract these coefficients, we need to subtract 2.3 from 6.8:

6.8 - 2.3 = 4.5

Step 4: Subtract the Exponents

The exponents are 7 and 7. Since the exponents are the same, we can subtract the coefficients and keep the same exponent:

4.5 × 10^7

Step 5: Write the Answer in Scientific Notation

The final answer is:

4.5 × 10^7

This is the answer in scientific notation.

Conclusion

In this article, we discussed the basics of scientific notation and how to calculate expressions involving large numbers. We used the expression (6.8×107)−(2.3×107)\left(6.8 \times 10^7\right) - \left(2.3 \times 10^7\right) as an example and followed the order of operations to evaluate the expression. The final answer is 4.5 × 10^7, which is the result of subtracting the two expressions in scientific notation.

Frequently Asked Questions

Q: What is scientific notation?

A: Scientific notation is a way of expressing very large or very small numbers in a more manageable form.

Q: How do I calculate expressions in scientific notation?

A: To calculate expressions in scientific notation, you need to follow the order of operations (PEMDAS).

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:

  1. Parentheses
  2. Exponents
  3. Multiplication and Division
  4. Addition and Subtraction

Q: How do I subtract expressions in scientific notation?

A: To subtract expressions in scientific notation, you need to subtract the coefficients and the exponents separately.

Q: What is the final answer to the expression (6.8×107)−(2.3×107)\left(6.8 \times 10^7\right) - \left(2.3 \times 10^7\right)?

A: The final answer is 4.5 × 10^7.

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In our previous article, we discussed the basics of scientific notation and how to calculate expressions involving large numbers. However, we know that there are many more questions that you may have about scientific notation. In this article, we will answer some of the most frequently asked questions about scientific notation.

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

A: Scientific notation and standard notation are two different ways of expressing numbers. Standard notation is the way we normally write numbers, with a decimal point and a whole number part. Scientific notation, on the other hand, 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 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 is the rule for multiplying numbers in scientific notation?

A: When multiplying numbers in scientific notation, you need to multiply the coefficients and add the exponents. For example, if you have the numbers 3.4 × 10^5 and 2.1 × 10^3, you would multiply the coefficients (3.4 and 2.1) and add the exponents (5 and 3).

Q: What is the rule for dividing numbers in scientific notation?

A: When dividing numbers in scientific notation, you need to divide the coefficients and subtract the exponents. For example, if you have the numbers 3.4 × 10^5 and 2.1 × 10^3, you would divide the coefficients (3.4 and 2.1) and subtract the exponents (5 and 3).

Q: How do I add or subtract numbers in scientific notation?

A: When adding or subtracting numbers in scientific notation, you need to have the same exponent for both numbers. If the exponents are different, you need to convert one of the numbers to have the same exponent as the other number.

Q: What is the rule for raising a number to a power in scientific notation?

A: When raising a number to a power in scientific notation, you need to raise the coefficient to the power and multiply the exponent by the power. For example, if you have the number 3.4 × 10^5 and you want to raise it to the power of 2, you would raise the coefficient (3.4) to the power of 2 and multiply the exponent (5) by 2.

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 coefficient by 10 raised to the power of the exponent. For example, if you have the number 3.4 × 10^5, you would multiply the coefficient (3.4) by 10 raised to the power of 5.

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:

  • Not following the order of operations (PEMDAS)
  • Not having the same exponent for both numbers when adding or subtracting
  • Not raising the coefficient to the correct power when raising a number to a power
  • Not converting numbers to have the same exponent when adding or subtracting

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

A: Scientific notation is used in many real-life situations, including:

  • Calculating distances and velocities in physics and engineering
  • Measuring the size of objects in astronomy and geology
  • Calculating the amount of a substance in chemistry and biology
  • Measuring the size of a population in statistics and demographics

Q: What are some common applications of scientific notation?

A: Some common applications of scientific notation include:

  • Calculating the distance to the moon or the sun
  • Measuring the size of a mountain or a valley
  • Calculating the amount of a substance in a chemical reaction
  • Measuring the size of a population in a city or a country

Q: How do I choose the correct exponent when working with scientific notation?

A: When choosing the correct exponent when working with scientific notation, you need to consider the magnitude of the number. If the number is very large, you will need to use a positive exponent. If the number is very small, you will need to use a negative exponent.

Q: What are some common misconceptions about scientific notation?

A: Some common misconceptions about scientific notation include:

  • Thinking that scientific notation is only used for very large numbers
  • Thinking that scientific notation is only used for mathematical calculations
  • Thinking that scientific notation is only used in science and engineering

Q: How do I use scientific notation to solve problems in mathematics and science?

A: To use scientific notation to solve problems in mathematics and science, you need to follow the order of operations (PEMDAS) and use the rules for multiplying, dividing, adding, and subtracting numbers in scientific notation. You also need to be able to convert numbers from standard notation to scientific notation and vice versa.

Q: What are some common challenges when working with scientific notation?

A: Some common challenges when working with scientific notation include:

  • Not following the order of operations (PEMDAS)
  • Not having the same exponent for both numbers when adding or subtracting
  • Not raising the coefficient to the correct power when raising a number to a power
  • Not converting numbers to have the same exponent when adding or subtracting

Q: How do I overcome common challenges when working with scientific notation?

A: To overcome common challenges when working with scientific notation, you need to:

  • Follow the order of operations (PEMDAS)
  • Make sure to have the same exponent for both numbers when adding or subtracting
  • Raise the coefficient to the correct power when raising a number to a power
  • Convert numbers to have the same exponent when adding or subtracting

Q: What are some common resources for learning about scientific notation?

A: Some common resources for learning about scientific notation include:

  • Textbooks and online resources for mathematics and science
  • Online tutorials and videos
  • Practice problems and worksheets
  • Online communities and forums for mathematics and science

Q: How do I use online resources to learn about scientific notation?

A: To use online resources to learn about scientific notation, you need to:

  • Search for online tutorials and videos
  • Practice problems and worksheets
  • Online communities and forums for mathematics and science
  • Online textbooks and resources for mathematics and science

Q: What are some common tips for learning about scientific notation?

A: Some common tips for learning about scientific notation include:

  • Practice, practice, practice
  • Use online resources and tutorials
  • Watch videos and online lectures
  • Join online communities and forums for mathematics and science
  • Read textbooks and online resources for mathematics and science

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

A: To use scientific notation in real-world applications, you need to:

  • Understand the concept of scientific notation and how to use it
  • Apply the rules for multiplying, dividing, adding, and subtracting numbers in scientific notation
  • Use scientific notation to solve problems in mathematics and science
  • Convert numbers from standard notation to scientific notation and vice versa

Q: What are some common applications of scientific notation in real-world scenarios?

A: Some common applications of scientific notation in real-world scenarios include:

  • Calculating distances and velocities in physics and engineering
  • Measuring the size of objects in astronomy and geology
  • Calculating the amount of a substance in chemistry and biology
  • Measuring the size of a population in statistics and demographics

Q: How do I choose the correct exponent when working with scientific notation in real-world applications?

A: When choosing the correct exponent when working with scientific notation in real-world applications, you need to consider the magnitude of the number. If the number is very large, you will need to use a positive exponent. If the number is very small, you will need to use a negative exponent.

Q: What are some common challenges when working with scientific notation in real-world applications?

A: Some common challenges when working with scientific notation in real-world applications include:

  • Not following the order of operations (PEMDAS)
  • Not having the same exponent for both numbers when adding or subtracting
  • Not raising the coefficient to the correct power when raising a number to a power
  • Not converting numbers to have the same exponent when adding or subtracting

Q: How do I overcome common challenges when working with scientific notation in real-world applications?

A: To overcome common challenges when working with scientific notation in real-world applications, you need to:

  • Follow the order of operations (PEMDAS)
  • Make sure to have the same exponent for both numbers when adding or subtracting
  • Raise the coefficient to the correct power when raising a number to a power
  • Convert numbers to have the same exponent when adding or subtracting

Q: What are some common resources for learning about scientific notation in real-world applications?

A: Some common resources for learning about scientific notation in real-world applications include:

  • Textbooks and online resources for mathematics and science
  • Online tutorials and videos
  • Practice problems and worksheets
  • Online communities and forums for mathematics and science

Q: How do I use online resources to learn about