Simplifying Scientific Notation (9.78 × 10^-11)(3.4 × 10^-18)

Introduction

Hey guys! Today, we're diving into the fascinating world of scientific notation and tackling a problem that might seem intimidating at first glance. But don't worry, we're going to break it down step by step and make it super easy to understand. Our mission is to simplify the expression (9.78 × 10^-11)(3.4 × 10^-18) and present the final answer in the correct scientific notation format. Scientific notation is a neat way to represent very large or very small numbers, making them easier to work with. It's used a lot in fields like science and engineering, so mastering it is a valuable skill. Let’s get started and demystify this mathematical puzzle!

Understanding Scientific Notation

Before we jump into the problem, let’s quickly recap what scientific notation actually is. In essence, scientific notation expresses a number as the product of two parts: a coefficient (a number between 1 and 10) and a power of 10. This format allows us to handle numbers that are either incredibly large or infinitesimally small with much more ease. For example, the number 3,000,000,000 can be written as 3 × 10^9, and the number 0.0000000007 can be expressed as 7 × 10^-10. This not only saves space but also simplifies calculations. When you're dealing with scientific notation, you're essentially managing the magnitude of the number separately from its significant digits. This separation is what makes it so powerful. Imagine trying to multiply or divide numbers with dozens of zeros – it would be a nightmare! But with scientific notation, you just multiply or divide the coefficients and add or subtract the exponents, making the whole process much more manageable. Understanding this fundamental concept is crucial for tackling our problem effectively. So, let's keep this in mind as we move forward and simplify the expression at hand. Remember, scientific notation is your friend when it comes to dealing with extreme numbers! It's all about breaking things down into manageable pieces.

Step-by-Step Simplification

Now, let’s get our hands dirty and simplify the expression (9.78 × 10^-11)(3.4 × 10^-18) step by step. This is where the magic happens! The key to simplifying expressions in scientific notation is to handle the coefficients and the powers of 10 separately. First, we multiply the coefficients: 9.78 multiplied by 3.4. Grab your calculators, guys, or do it the old-fashioned way – whatever floats your boat! The result of this multiplication is 33.252. Now, let’s tackle the powers of 10. We have 10^-11 and 10^-18. When you multiply numbers with the same base, you simply add the exponents. So, we add -11 and -18, which gives us -29. This means we have 10^-29. Putting these two results together, we get 33.252 × 10^-29. But wait! We're not quite done yet. Remember, scientific notation requires the coefficient to be between 1 and 10. Our coefficient, 33.252, is a bit too big. So, we need to adjust it. We can rewrite 33.252 as 3.3252 × 10^1. Now, we have (3.3252 × 10^1) × 10^-29. Again, we add the exponents: 1 + (-29) = -28. Voila! Our final answer in scientific notation is 3.3252 × 10^-28. See? It's not so scary when you break it down. We multiplied the coefficients, added the exponents, and adjusted the coefficient to fit the scientific notation format. You've nailed it!

Detailed Solution

Let's break down the detailed solution for simplifying the expression (9.78 × 10^-11)(3.4 × 10^-18), so every step is crystal clear. This way, you’ll not only get the right answer but also understand the why behind each move. First up, we focus on the coefficients. We need to multiply 9.78 by 3.4. This is a straightforward multiplication, and when you crunch the numbers, you get 33.252. Easy peasy! Next, we turn our attention to the powers of 10. We have 10^-11 and 10^-18. Here’s where the exponent rules come in handy. When multiplying exponential terms with the same base, you simply add the exponents. So, we add -11 and -18, resulting in -29. This gives us 10^-29. Now, we combine the results from our coefficient and exponent calculations. We have 33.252 × 10^-29. Almost there, but not quite in proper scientific notation form yet! Remember, the coefficient needs to be a number between 1 and 10. 33.252 is larger than 10, so we need to adjust it. We can rewrite 33.252 as 3.3252 × 10^1. Think of it as moving the decimal point one place to the left, which increases the power of 10 by one. Now our expression looks like this: (3.3252 × 10^1) × 10^-29. One last step! We need to combine the powers of 10 again. We add the exponents 1 and -29, which gives us -28. So, the final simplified expression in scientific notation is 3.3252 × 10^-28. Boom! You’ve done it. We walked through each step, from multiplying the coefficients to adjusting the final answer into the correct format. Understanding these steps will make tackling similar problems a breeze. You're becoming a scientific notation pro!

Writing the Final Answer in Scientific Notation

Alright, let's talk about writing the final answer in scientific notation. We've already done the hard work of simplifying the expression, but presenting it in the correct format is crucial. Remember, the key to scientific notation is to express a number as a product of a coefficient and a power of 10, where the coefficient is a number between 1 and 10. We've arrived at the intermediate result of 33.252 × 10^-29. This is mathematically correct, but it's not in proper scientific notation because 33.252 is greater than 10. So, what do we do? We need to adjust the coefficient to be within the 1 to 10 range. We can do this by moving the decimal point one place to the left, which means we rewrite 33.252 as 3.3252. But we can’t just change the number without compensating for it. When we decrease the coefficient by a factor of 10 (moving the decimal one place to the left), we need to increase the exponent by 1 to keep the overall value the same. This is where the power of exponents shines! So, we rewrite 33.252 × 10^-29 as 3.3252 × 10^1 × 10^-29. Now, we combine the powers of 10 by adding the exponents: 1 + (-29) = -28. This gives us 10^-28. Finally, we can write our answer in perfect scientific notation: 3.3252 × 10^-28. This format clearly shows the magnitude and significant figures of the number. The coefficient, 3.3252, tells us the precise value, and the exponent, -28, tells us how many places to move the decimal point to get the number in standard form (which would be a very, very small number in this case). Presenting your answer in scientific notation not only shows that you've solved the problem correctly but also demonstrates a clear understanding of how numbers are represented in this format. It's a clean, efficient, and widely accepted way to express very large or very small values. So, pat yourselves on the back – you’ve mastered another essential math skill!

Common Mistakes to Avoid

Let’s chat about some common pitfalls to sidestep when you're simplifying expressions in scientific notation. Knowing these mistakes can save you a lot of headaches and ensure you nail the correct answer every time. One of the most frequent errors is forgetting to adjust the coefficient after multiplying. Remember, the coefficient must always be between 1 and 10. If you end up with a coefficient that’s greater than 10 or less than 1, you need to tweak it and adjust the exponent accordingly. For example, if you get 33.252 × 10^-29, you can't just leave it like that. You have to convert it to 3.3252 × 10^-28. Another common mistake is messing up the exponent rules. When multiplying numbers in scientific notation, you add the exponents, and when dividing, you subtract them. It’s easy to mix these up, so always double-check your operations. A simple way to remember is that multiplication and addition go together, as do division and subtraction. Sign errors are another sneaky culprit. Make sure you’re paying close attention to negative signs, especially when adding or subtracting exponents. For instance, -11 + (-18) is -29, not -7. Keeping track of these signs is crucial for getting the correct exponent. Lastly, some people forget to rewrite the final answer in proper scientific notation. You might correctly calculate the coefficient and the power of 10, but if you leave the answer as, say, 0.5 × 10^-27, it’s not in the correct format. You’d need to adjust it to 5 × 10^-28. Avoiding these common mistakes is all about paying attention to detail and double-checking your work. Practice makes perfect, so the more you work with scientific notation, the more natural these adjustments will become. You’ll be spotting these errors like a pro in no time!

Real-World Applications of Scientific Notation

Okay, guys, let's step back for a moment and appreciate why we're even learning about scientific notation in the first place. It's not just some abstract math concept – it's a powerful tool that's used extensively in the real world, especially in science and engineering. Think about it: we often deal with numbers that are either incredibly huge or unbelievably tiny. For instance, the speed of light is approximately 300,000,000 meters per second, and the size of an atom is around 0.0000000001 meters. Writing these numbers out in full can be cumbersome and prone to errors. That's where scientific notation comes to the rescue! In astronomy, the distances between stars and galaxies are so vast that they're almost incomprehensible. Scientists use scientific notation to express these distances in a manageable way. For example, the distance to the Andromeda Galaxy is roughly 2.5 × 10^22 meters. Imagine trying to write that out with all the zeros! Chemistry and physics also rely heavily on scientific notation. The Avogadro constant, which represents the number of atoms or molecules in a mole of a substance, is approximately 6.022 × 10^23. Similarly, the Planck constant, a fundamental constant in quantum mechanics, is about 6.626 × 10^-34 joule-seconds. These constants are crucial for calculations in these fields, and scientific notation makes them much easier to work with. Even in computer science, scientific notation plays a role. The storage capacity of hard drives and the processing speed of computers are often expressed using powers of 10, which can be conveniently written in scientific notation. So, the next time you're struggling with a scientific notation problem, remember that you're learning a skill that's used by scientists, engineers, and other professionals every day. It's not just about the math – it's about having a tool that helps you understand and describe the world around you. Pretty cool, huh?

Conclusion

So, there you have it, guys! We've successfully simplified the expression (9.78 × 10^-11)(3.4 × 10^-18) and written the final answer in scientific notation. We walked through the entire process step by step, from multiplying the coefficients to adjusting the final result into the correct format. Remember, the key to mastering scientific notation is to break down the problem into smaller, manageable parts. Multiply the coefficients, add the exponents, and make sure your final answer is in the proper format, with a coefficient between 1 and 10. We also discussed some common mistakes to avoid, like forgetting to adjust the coefficient or messing up the exponent rules. Keeping these pitfalls in mind will help you solve these problems with confidence. And we explored the real-world applications of scientific notation, showing how this powerful tool is used in various fields like astronomy, chemistry, physics, and computer science. It’s not just about the math; it’s about understanding and describing the world around us. By understanding the principles and practicing regularly, you’ll become a scientific notation whiz in no time. Keep up the great work, and remember, math can be fun when you break it down and tackle it step by step! Now you're equipped to handle similar problems with ease. Go forth and conquer those scientific notations!