Calculating (7 X 10^2) X (6 X 10^3): A Math Solution

by SLV Team 53 views
Calculating (7 x 10^2) x (6 x 10^3): A Math Solution

Let's break down how to solve this mathematical problem. This article will guide you through the steps to correctly calculate the result of (7 x 10^2) x (6 x 10^3) and help you understand the underlying principles of scientific notation.

Understanding Scientific Notation

Before diving into the calculation, it's crucial to understand scientific notation. Scientific notation is a way of expressing numbers that are either very large or very small in a compact and convenient form. It's written as a number between 1 and 10 (the coefficient) multiplied by a power of 10. For example, 300,000,000 can be written as 3 x 10^8 in scientific notation. This notation makes it easier to handle large numbers in calculations and comparisons.

In our problem, we have two numbers expressed in scientific notation: 7 x 10^2 and 6 x 10^3. The exponents (2 and 3) indicate the power of 10 that the coefficients (7 and 6) are multiplied by. Essentially, 7 x 10^2 is 7 multiplied by 10 raised to the power of 2 (which is 100), and 6 x 10^3 is 6 multiplied by 10 raised to the power of 3 (which is 1000). Mastering scientific notation is an important step in math, especially for complex calculations. It is also a great way to present huge numbers on your paper or screen without using a ton of space.

Step-by-Step Calculation

Now, let's calculate the result of (7 x 10^2) x (6 x 10^3). To do this, we can use the associative and commutative properties of multiplication, which allow us to rearrange the terms and group them in a way that makes the calculation easier. The equation can be rewritten to group the numbers and the powers of 10 separately. This makes the calculation process smoother and less prone to errors. Let's walk through each step to ensure clarity and understanding.

  1. Rearrange the terms:

    (7 x 10^2) x (6 x 10^3) can be rearranged as (7 x 6) x (10^2 x 10^3).

    This step is crucial because it separates the coefficients (7 and 6) from the powers of 10 (10^2 and 10^3), making the subsequent multiplication easier to manage.

  2. Multiply the coefficients:

    Multiply 7 by 6: 7 x 6 = 42.

    This is a straightforward multiplication step. The result, 42, will be the coefficient of our final answer before we adjust it for scientific notation if needed.

  3. Multiply the powers of 10:

    To multiply 10^2 by 10^3, we use the rule of exponents that states a^m x a^n = a^(m+n). So, 10^2 x 10^3 = 10^(2+3) = 10^5.

    This step utilizes a fundamental property of exponents. When multiplying numbers with the same base (in this case, 10), you add the exponents. This simplifies the multiplication of powers of 10 significantly.

  4. Combine the results:

    Now, combine the results from steps 2 and 3: 42 x 10^5.

    This step brings together the product of the coefficients and the product of the powers of 10, giving us an intermediate result in scientific notation form.

  5. Adjust for proper scientific notation:

    In proper scientific notation, the coefficient should be between 1 and 10. 42 is greater than 10, so we need to rewrite it as 4.2 x 10^1. Therefore, 42 x 10^5 becomes (4.2 x 10^1) x 10^5.

    This is a critical step for expressing the answer in the correct scientific notation format. By ensuring the coefficient is between 1 and 10, we adhere to the standard conventions of scientific notation.

  6. Simplify the exponents:

    Using the same rule of exponents as before, 10^1 x 10^5 = 10^(1+5) = 10^6.

    This step simplifies the expression further by combining the powers of 10, making the final result clearer.

  7. Final result:

    The final result is 4.2 x 10^6.

    This is the solution to our problem, expressed in proper scientific notation. It represents the product of the original numbers in a concise and easily understandable form.

Identifying the Correct Option

Looking at the given options:

  • igcirc 13 x 10^-1
  • igcirc 4.2 x 10^-1
  • igcirc 4.2 x 10^5
  • igcirc 13 x 10^5
  • igcirc 4.2 x 10^6

The correct answer is igcirc 4.2 x 10^6.

This matches the result we calculated in the previous section. Understanding how to work with scientific notation is super important for solving these kinds of problems.

Common Mistakes to Avoid

When working with scientific notation, there are a few common mistakes that students often make. Avoiding these pitfalls can help ensure accurate calculations and a better understanding of the concepts. Let's go through some of the common errors and how to prevent them.

  1. Incorrectly applying the exponent rule:

    One common mistake is misapplying the rule for multiplying powers with the same base. Remember that when multiplying a^m by a^n, you add the exponents (a^m x a^n = a^(m+n)). For instance, when multiplying 10^2 by 10^3, you should add the exponents to get 10^(2+3) = 10^5, not multiply them. Students sometimes mistakenly multiply the exponents, which leads to an incorrect result.

    • How to avoid it: Always double-check that you are adding the exponents when multiplying powers with the same base. Writing out the rule (a^m x a^n = a^(m+n)) can serve as a helpful reminder during your calculations. Practice with different examples to reinforce your understanding of this rule.

  2. Forgetting to adjust the coefficient:

    Another frequent error is forgetting to adjust the coefficient to be between 1 and 10 when expressing the final answer in scientific notation. If your calculation results in a coefficient that is greater than 10 or less than 1, you need to adjust it accordingly. For example, if you get 42 x 10^5 as an intermediate result, you need to rewrite 42 as 4.2 x 10^1 and then combine the powers of 10 to get 4.2 x 10^6.

    • How to avoid it: Always review your final answer to ensure that the coefficient is between 1 and 10. If it's not, adjust the coefficient and the exponent accordingly. This might involve moving the decimal point and updating the exponent to maintain the correct value. Regularly check this to ensure accuracy.

  3. Misunderstanding negative exponents:

    Negative exponents can also cause confusion. A negative exponent indicates a number less than 1, specifically the reciprocal of the base raised to the positive exponent. For example, 10^-2 is equal to 1/10^2, which is 0.01. Misinterpreting negative exponents can lead to significant errors in calculations.

    • How to avoid it: Take the time to understand what negative exponents mean. Remember that a negative exponent signifies a fraction. If you encounter a negative exponent, rewrite it as a reciprocal (e.g., 10^-n = 1/10^n) before proceeding with the calculation. This simple step can prevent many mistakes. A good grasp of this will definitely help with scientific notation.

  4. Mixing up addition/subtraction with multiplication/division:

    Students sometimes confuse the rules for adding and subtracting numbers in scientific notation with the rules for multiplication and division. When adding or subtracting, the numbers must have the same power of 10, while multiplication and division involve different rules for handling exponents.

    • How to avoid it: Keep the rules for each operation separate in your mind. When multiplying or dividing, focus on adding or subtracting the exponents, respectively. When adding or subtracting, make sure the powers of 10 are the same before performing the operation. A clear distinction between these operations is essential for accuracy.

By being aware of these common mistakes and actively working to avoid them, you can improve your accuracy and confidence when working with scientific notation.

Practice Problems

To solidify your understanding of scientific notation and calculations involving it, here are a few practice problems. Working through these problems will help you become more comfortable with the concepts and techniques we've discussed. Practice makes perfect, and these problems will provide valuable hands-on experience.

  1. (3 x 10^4) x (2 x 10^2) = ?

    • This problem is similar to the one we worked through in the article. Try to apply the same steps: rearrange the terms, multiply the coefficients, multiply the powers of 10, and adjust for proper scientific notation if necessary. This exercise will reinforce the basic principles of multiplying numbers in scientific notation.

  2. (8 x 10^6) / (4 x 10^3) = ?

    • This problem involves division. Remember that when dividing numbers in scientific notation, you divide the coefficients and subtract the exponents. Pay close attention to the exponent subtraction to ensure you get the correct power of 10 in your answer. Mastering division in scientific notation is crucial for a complete understanding.

  3. (5 x 10^-3) x (7 x 10^5) = ?

    • This problem includes a negative exponent. Be sure to handle the negative exponent correctly by adding it to the other exponent during multiplication. This exercise will help you practice working with both positive and negative exponents, a key aspect of scientific notation.

  4. (9 x 10^2) + (3 x 10^2) = ?

    • This problem involves addition. Remember that you can only add numbers in scientific notation if they have the same power of 10. Since these numbers already have the same power of 10, you can simply add the coefficients. If the powers of 10 were different, you would need to adjust one of the numbers to match the other before adding. Addition and subtraction in scientific notation are often overlooked but equally important.

Conclusion

Calculating (7 x 10^2) x (6 x 10^3) involves understanding scientific notation and applying the rules of exponents. The correct answer is 4.2 x 10^6. Remember to practice these steps to improve your math skills! You've got this, guys! Keep up the great work, and you'll be a math whiz in no time!