Scientific Notation Calculations: Step-by-Step Solutions

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Hey guys! Are you struggling with scientific notation? Don't worry, you're not alone! Scientific notation can seem tricky at first, but once you understand the basics, it's actually quite straightforward. In this article, we'll break down how to solve problems involving scientific notation, step by step. We'll tackle four examples that cover different scenarios, so you'll be well-equipped to handle any scientific notation calculation that comes your way. Let's dive in!

Understanding Scientific Notation

Before we jump into solving problems, let's quickly recap what scientific notation is and why it's so useful. Scientific notation is a way of expressing very large or very small numbers in a compact and standardized form. It's especially handy in fields like science and engineering where you often deal with numbers that have many digits. The general form of scientific notation is a × 10^b, where:

  • 1 ≤ |a| < 10 (The absolute value of a is between 1 and 10, including 1 but excluding 10)
  • a is the coefficient or significand
  • b is an integer exponent

Think of it this way: scientific notation allows us to write numbers as a product of a number between 1 and 10 and a power of 10. This makes it easier to compare and manipulate very large or small numbers. For example, instead of writing 0.0000000056, we can write 5.6 × 10⁻⁹. Similarly, instead of writing 9,200,000,000, we can write 9.2 × 10⁹. See how much simpler that is?

Why is this so important? Imagine you're a physicist working with the mass of an electron or the distance to a distant galaxy. These numbers are incredibly small and large, respectively. Writing them out in their full form would be cumbersome and prone to errors. Scientific notation provides a neat and efficient way to handle these numbers, making calculations and comparisons much easier. Plus, it helps us keep track of significant figures, which is crucial in scientific measurements. So, understanding scientific notation is a fundamental skill for anyone working with numbers in a scientific context. Now that we've refreshed our understanding of what scientific notation is, let's get into solving some problems!

Problem A: 8.2 * 10² * 9 * 10³

Let's start with the first problem: 8.2 * 10² * 9 * 10³. This problem involves multiplying two numbers expressed in scientific notation. To solve this, we'll use the basic rules of exponents and multiplication. Remember, when multiplying exponents with the same base, we add the powers. Here's how we can break it down step-by-step:

  1. Group the coefficients and the powers of 10: The first step is to rearrange the terms so that the coefficients (8.2 and 9) are together and the powers of 10 (10² and 10³) are together. This makes the multiplication process clearer. We can rewrite the expression as (8.2 * 9) * (10² * 10³).
  2. Multiply the coefficients: Now, let's multiply the coefficients: 8.2 * 9 = 73.8. This gives us the numerical part of our answer.
  3. Multiply the powers of 10: Next, we multiply the powers of 10. Remember the rule: when multiplying exponents with the same base, we add the powers. So, 10² * 10³ = 10^(2+3) = 10⁵. This means we have 10 raised to the power of 5.
  4. Combine the results: Now, we combine the results from steps 2 and 3. We have 73.8 * 10⁵.
  5. Adjust to scientific notation: The final step is crucial. Remember, in scientific notation, the coefficient must be between 1 and 10 (excluding 10). Our current coefficient is 73.8, which is greater than 10. To fix this, we need to move the decimal point one place to the left, making the coefficient 7.38. When we do this, we're essentially dividing by 10, so we need to multiply the power of 10 by 10 to compensate. This means we increase the exponent by 1. So, 73.8 * 10⁵ becomes 7.38 * 10^(5+1) = 7.38 * 10⁶.

Therefore, the answer to 8.2 * 10² * 9 * 10³ in scientific notation is 7.38 * 10⁶. See how we broke it down into manageable steps? By grouping the coefficients and powers of 10, multiplying them separately, and then adjusting the result to fit the scientific notation format, we arrived at the correct answer. This systematic approach will help you tackle similar problems with confidence. Let's move on to the next problem!

Problem B: 3.7 * 10⁵ * 8.6 * 10³

Okay, let's move on to the next problem: 3.7 * 10⁵ * 8.6 * 10³. This one is similar to the previous problem, so we'll use the same steps to solve it. It's all about practice, guys! The more you do these, the easier they become. Let's break it down:

  1. Group the coefficients and the powers of 10: Just like before, we'll group the coefficients (3.7 and 8.6) together and the powers of 10 (10⁵ and 10³) together. This gives us (3.7 * 8.6) * (10⁵ * 10³).
  2. Multiply the coefficients: Let's multiply the coefficients: 3.7 * 8.6 = 31.82. This is the numerical part of our answer.
  3. Multiply the powers of 10: Now, we multiply the powers of 10. Again, we add the exponents: 10⁵ * 10³ = 10^(5+3) = 10⁸. We now have 10 raised to the power of 8.
  4. Combine the results: We combine the results from steps 2 and 3, giving us 31.82 * 10⁸.
  5. Adjust to scientific notation: Remember, the coefficient needs to be between 1 and 10. Our coefficient, 31.82, is larger than 10. So, we move the decimal point one place to the left, making the coefficient 3.182. Since we divided by 10, we need to multiply the power of 10 by 10 to compensate, increasing the exponent by 1. Thus, 31.82 * 10⁸ becomes 3.182 * 10^(8+1) = 3.182 * 10⁹.

So, the answer to 3.7 * 10⁵ * 8.6 * 10³ in scientific notation is 3.182 * 10⁹. Did you follow along? Notice how the steps are the same as in the previous problem? This is the beauty of scientific notation – once you get the method, you can apply it to various problems. Let's keep the momentum going and tackle the next one!

Problem C: 3.95 * 10⁸ * 6.74 * 10⁻⁹

Alright, let's jump into problem C: 3.95 * 10⁸ * 6.74 * 10⁻⁹. This one introduces a negative exponent, but don't let that scare you! The same principles apply. We'll just need to be careful when adding the exponents. Let’s break it down, step by step, just like we've been doing. You'll see, negative exponents are no match for our problem-solving skills!

  1. Group the coefficients and the powers of 10: As always, we start by grouping the coefficients (3.95 and 6.74) and the powers of 10 (10⁸ and 10⁻⁹). This gives us (3.95 * 6.74) * (10⁸ * 10⁻⁹).
  2. Multiply the coefficients: Now, let's multiply those coefficients: 3.95 * 6.74 = 26.623. We've got our numerical part.
  3. Multiply the powers of 10: Here’s where we handle the negative exponent. When multiplying powers of 10, we add the exponents: 10⁸ * 10⁻⁹ = 10^(8 + (-9)) = 10^(8 - 9) = 10⁻¹. So, we end up with 10 to the power of -1.
  4. Combine the results: Let's put it all together: 26.623 * 10⁻¹.
  5. Adjust to scientific notation: Our coefficient, 26.623, is greater than 10, so we need to adjust it. We move the decimal point one place to the left, making the coefficient 2.6623. Because we divided by 10, we need to compensate by multiplying the power of 10 by 10. This means adding 1 to the exponent: -1 + 1 = 0. So, 26.623 * 10⁻¹ becomes 2.6623 * 10⁰.

Therefore, the answer to 3.95 * 10⁸ * 6.74 * 10⁻⁹ in scientific notation is 2.6623 * 10⁰. Remember that 10⁰ is equal to 1, so we could also write this as simply 2.6623. See, even with a negative exponent, the process is the same. The key is to take it one step at a time and remember the rules of exponents. Let's tackle the final problem!

Problem D: 9.7 * 10⁻² * 5.7 * 10⁻⁶

Okay, guys, let's tackle the last problem: 9.7 * 10⁻² * 5.7 * 10⁻⁶. This one has two negative exponents, so it's a great opportunity to solidify our understanding. We'll follow the same steps we've been using, and you'll see that handling multiple negative exponents is no different. Let's break it down and conquer this final problem!

  1. Group the coefficients and the powers of 10: As we've consistently done, we group the coefficients (9.7 and 5.7) and the powers of 10 (10⁻² and 10⁻⁶). This gives us (9.7 * 5.7) * (10⁻² * 10⁻⁶).
  2. Multiply the coefficients: Let's multiply those coefficients: 9.7 * 5.7 = 55.29. We've got our numerical part.
  3. Multiply the powers of 10: Now, let's deal with the powers of 10. Remember, we add the exponents: 10⁻² * 10⁻⁶ = 10^(-2 + (-6)) = 10^(-2 - 6) = 10⁻⁸. So, we end up with 10 to the power of -8.
  4. Combine the results: Let's put it all together: 55.29 * 10⁻⁸.
  5. Adjust to scientific notation: Our coefficient, 55.29, is greater than 10, so we need to adjust it. We move the decimal point one place to the left, making the coefficient 5.529. Because we divided by 10, we need to compensate by multiplying the power of 10 by 10. This means adding 1 to the exponent: -8 + 1 = -7. So, 55.29 * 10⁻⁸ becomes 5.529 * 10⁻⁷.

Therefore, the answer to 9.7 * 10⁻² * 5.7 * 10⁻⁶ in scientific notation is 5.529 * 10⁻⁷. Fantastic! We've solved all four problems. You've seen how the same method applies, even when dealing with negative exponents. The key is to break the problem down into steps, focus on one step at a time, and remember the rules of exponents.

Conclusion

So, there you have it! We've walked through four different scientific notation problems, covering various scenarios. You've learned how to multiply numbers in scientific notation, how to handle positive and negative exponents, and how to adjust your answers to fit the standard scientific notation format. Remember, the key to mastering scientific notation is practice. Work through more problems, and you'll become more comfortable with the process. Scientific notation is a powerful tool in science and mathematics, and with a solid understanding, you'll be well-equipped to tackle any numerical challenge that comes your way. Keep practicing, and you'll become a scientific notation pro in no time! You got this!