Solving (3.1 X 10^5)(2.2 X 10^7) In Scientific Notation

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Before diving into the solution, let's first understand what scientific notation is and why it's used. Scientific notation is a standardized way of expressing very large or very small numbers. It's particularly useful in scientific and mathematical contexts where dealing with numbers that have many digits is common. Scientific notation expresses a number as a product of two parts: a coefficient and a power of 10. The coefficient is a decimal number between 1 (inclusive) and 10 (exclusive), and the power of 10 indicates the number's magnitude.

For example, the number 300,000,000 can be written in scientific notation as 3 x 10^8. Here, 3 is the coefficient, and 10^8 represents 10 raised to the power of 8, indicating that the decimal point in the coefficient should be moved 8 places to the right to obtain the original number. Similarly, a small number like 0.00000025 can be written as 2.5 x 10^-7, where the negative exponent indicates that the decimal point should be moved 7 places to the left.

The key advantage of using scientific notation is its ability to simplify the representation and manipulation of numbers, especially in calculations involving very large or very small values. It also makes it easier to compare the relative magnitudes of different numbers. In this article, we'll use the principles of scientific notation to solve the given problem and determine the correct answer from the provided options.

Now, let's focus on the problem at hand: (3.1 x 10^5)(2.2 x 10^7). This problem requires us to multiply two numbers expressed in scientific notation. The numbers are 3.1 x 10^5 and 2.2 x 10^7. To solve this, we will apply the basic principles of multiplication and exponent rules. The primary goal is to obtain the result in scientific notation, which means the final answer should also be in the form of a coefficient multiplied by a power of 10.

Multiplying numbers in scientific notation involves two main steps: First, we multiply the coefficients (the decimal numbers). In this case, we multiply 3.1 by 2.2. Second, we multiply the powers of 10. Here, we multiply 10^5 by 10^7. When multiplying powers with the same base (in this case, 10), we add the exponents. So, 10^5 multiplied by 10^7 becomes 10^(5+7), which is 10^12. After performing these two steps, we combine the results to get the product in scientific notation. However, it's important to check if the resulting coefficient is between 1 and 10. If it's not, we need to adjust it and the exponent accordingly to maintain the correct scientific notation format.

To solve the expression (3.1 x 10^5)(2.2 x 10^7), we will break down the calculation into manageable steps. This methodical approach ensures accuracy and clarity in the solution. The process involves multiplying the coefficients and then multiplying the powers of 10, followed by adjusting the result to fit the standard scientific notation format. By carefully following each step, we can confidently arrive at the correct answer.

  1. Multiply the coefficients: The first step is to multiply the coefficients, which are the decimal numbers in front of the powers of 10. In this case, we need to multiply 3.1 by 2.2.

    1. 1 x 2.2 = 6.82

    This multiplication is straightforward and results in 6.82.

  2. Multiply the powers of 10: Next, we multiply the powers of 10. We have 10^5 multiplied by 10^7. When multiplying powers with the same base, we add the exponents.

    10^5 x 10^7 = 10^(5+7) = 10^12

    So, the result of multiplying the powers of 10 is 10 raised to the power of 12.

  3. Combine the results: Now, we combine the results from the previous two steps. We have the product of the coefficients (6.82) and the product of the powers of 10 (10^12).

    1. 82 x 10^12

    This gives us 6.82 x 10^12.

  4. Check for scientific notation format: The final step is to ensure that our answer is in proper scientific notation. Scientific notation requires the coefficient to be a number between 1 and 10 (excluding 10). In our case, the coefficient is 6.82, which falls within this range. Therefore, the number is already in the correct scientific notation format.

After performing the multiplication and ensuring the result is in scientific notation, we have arrived at the final answer: 6.82 x 10^12. Now, let's compare this result with the options provided in the problem statement.

The options are:

  • A. 5.5 x 10^12
  • B. 6.82 x 10^12
  • C. 5.5 x 10^35
  • D. 6.82 x 10^35

By comparing our calculated answer with the options, it is clear that option B, 6.82 x 10^12, matches our result exactly. The other options have either different coefficients or different exponents, making them incorrect. Therefore, the correct answer to the problem (3.1 x 10^5)(2.2 x 10^7) in scientific notation is 6.82 x 10^12.

When working with scientific notation, it's easy to make mistakes if you're not careful. Understanding common errors can help prevent them and ensure accurate calculations. Here are some of the most frequent mistakes people make when dealing with scientific notation:

  1. Incorrectly Multiplying Coefficients: One common mistake is performing the multiplication of the coefficients incorrectly. It's crucial to accurately multiply the decimal numbers (in this case, 3.1 and 2.2). A simple arithmetic error here can lead to a completely wrong answer.

  2. Forgetting to Add Exponents: When multiplying numbers in scientific notation, you need to add the exponents of the powers of 10. A frequent mistake is either forgetting to add the exponents or adding them incorrectly. Remember the rule: when multiplying powers with the same base, you add the exponents (e.g., 10^5 * 10^7 = 10^(5+7)).

  3. Incorrectly Adjusting Scientific Notation Format: Scientific notation requires the coefficient to be a number between 1 and 10. If the coefficient you calculate is not within this range, you need to adjust it and modify the exponent accordingly. For instance, if you end up with 68.2 x 10^11, you need to adjust it to 6.82 x 10^12. Failing to make this adjustment is a common error.

  4. Misunderstanding Negative Exponents: Negative exponents in scientific notation represent numbers less than 1. For example, 10^-3 means 0.001. Misinterpreting negative exponents can lead to incorrect calculations and answers.

  5. Mixing Up Multiplication and Addition: It's essential to remember that when multiplying numbers in scientific notation, you multiply the coefficients and add the exponents. Some people mistakenly add the coefficients or multiply the exponents, which will result in an incorrect answer.

By being aware of these common pitfalls and carefully reviewing each step of your calculations, you can minimize the chances of making mistakes when working with scientific notation.

In conclusion, to solve the problem (3.1 x 10^5)(2.2 x 10^7), we followed the principles of scientific notation. We multiplied the coefficients (3.1 and 2.2) and added the exponents of the powers of 10 (5 and 7). This gave us a preliminary result of 6.82 x 10^12. We then verified that the result was in the correct scientific notation format, where the coefficient is a number between 1 and 10. Comparing our result with the given options, we confirmed that the correct answer is B. 6.82 x 10^12.

Understanding and applying scientific notation is crucial in various fields, including science, mathematics, and engineering. It simplifies the representation and manipulation of very large and very small numbers, making complex calculations more manageable. By following the steps outlined in this article and avoiding common mistakes, you can confidently work with scientific notation in a variety of contexts. This skill is not only valuable for academic purposes but also for real-world applications where dealing with large or small quantities is essential.