Multiplying Numbers In Scientific Notation A Step-by-Step Guide

Hey guys! Today, let's dive into multiplying numbers expressed in scientific notation. It might sound intimidating, but trust me, it's super straightforward once you get the hang of it. We'll break down a problem step by step, so you’ll be a pro in no time. Let's tackle this: Multiply $(4.21 imes 10^{-9})(7.8 imes 10^{15})$. We'll walk through each stage to ensure the final answer is in proper scientific notation.

Understanding Scientific Notation

Before we jump into the multiplication, let’s quickly recap what scientific notation actually is. Scientific notation is a way of expressing numbers, especially very large or very small numbers, in a more compact and readable form. It's written as a product of two parts: a coefficient (a number between 1 and 10) and a power of 10. For example, $3,000,000$ can be written as $3 imes 10^6$, and $0.000002$ can be written as $2 imes 10^{-6}$. This notation is incredibly useful in various fields like science and engineering, where dealing with such extreme numbers is common.

In scientific notation, the general form is $a imes 10^b$, where $1 imes 10^0$ ≤ |a| < 10 and b is an integer. The coefficient a provides the significant digits of the number, while the exponent b indicates the magnitude. A positive exponent means the original number was large (greater than 1), and a negative exponent means the original number was small (less than 1). This standardized format makes it easier to compare numbers and perform calculations. So, when we encounter scientific notation, we're essentially dealing with a shorthand that simplifies our calculations and expressions. Now that we've refreshed the basics, let’s move on to how we can actually multiply these numbers.

Step-by-Step Multiplication

Now, let's get to the main event: multiplying our numbers. We have $(4.21 imes 10^{-9})(7.8 imes 10^{15})$. The first step is to multiply the coefficients (the numbers in front of the powers of 10). So, we multiply $4.21$ by $7.8$. Grab your calculator or do it by hand, and you’ll find that $4.21 imes 7.8 = 32.838$. Keep this number in mind, as it will be part of our final answer. Next up, we need to deal with the powers of 10. When multiplying numbers in scientific notation, you multiply the coefficients and add the exponents. In our case, we have $10^{-9}$ and $10^{15}$. To combine these, we add the exponents: $-9 + 15 = 6$. So, we get $10^6$ as our combined power of 10. At this point, we have $32.838 imes 10^6$. This is technically the result of our multiplication, but it's not quite in proper scientific notation yet. Remember, the coefficient needs to be a number between 1 and 10. Our coefficient is $32.838$, which is larger than 10, so we need to adjust it. This adjustment is the key to getting our final answer in the correct form. Now, let's move on to the next step to make this adjustment and complete our problem.

Adjusting to Proper Scientific Notation

So, we’ve got $32.838 imes 10^6$, but remember, to be in proper scientific notation, that first number (the coefficient) needs to be between 1 and 10. $32.838$ is too big, so we need to make it smaller. We can do this by moving the decimal point one place to the left, making it $3.2838$. But here’s the catch: when we make the coefficient smaller, we need to make the exponent larger to keep the number the same. Since we moved the decimal point one place to the left, we increase the exponent by 1. So, $10^6$ becomes $10^{6+1}$, which is $10^7$. Think of it like balancing a seesaw: if you take something away from one side, you need to add something to the other to keep it balanced. In this case, we made the coefficient smaller by dividing by 10, so we make the exponent larger by multiplying by 10 (adding 1 to the exponent). Now, we have $3.2838 imes 10^7$. This is our number in proper scientific notation. The coefficient, $3.2838$, is between 1 and 10, and the exponent, 7, is an integer. We’ve successfully adjusted our result into the correct format. Now, let’s put it all together and see our final answer.

The Final Answer

Okay, let's recap what we've done. We started with $(4.21 imes 10^{-9})(7.8 imes 10^{15})$. We multiplied the coefficients $4.21$ and $7.8$ to get $32.838$. Then, we added the exponents $-9$ and $15$ to get $10^6$. This gave us $32.838 imes 10^6$. But, we needed to adjust this to proper scientific notation. We moved the decimal point in $32.838$ one place to the left to get $3.2838$, and we increased the exponent by 1 to get $10^7$. So, our final answer is $3.2838 imes 10^7$. This is our result, expressed perfectly in scientific notation. You see, it wasn’t so bad, right? By breaking it down step by step, we made the process manageable. Remember, the key is to multiply the coefficients, add the exponents, and then adjust the result to fit the standard scientific notation format. Now that you've mastered this, you'll be able to handle similar problems with ease. Let's do a quick review of the key concepts to really solidify your understanding.

Reviewing the Key Concepts

Let’s quickly run through the key concepts we covered today, just to make sure everything is crystal clear. First, scientific notation is a way to express numbers as a coefficient multiplied by a power of 10. This is super helpful for dealing with very large or very small numbers. The general form is $a imes 10^b$, where $1 imes 10^0$ ≤ |a| < 10 and b is an integer. When you multiply numbers in scientific notation, you multiply the coefficients and add the exponents. For example, if you have $(a imes 10^m)(b imes 10^n)$, you multiply a and b, and you add m and n. Finally, and this is crucial, make sure your answer is in proper scientific notation. This means the coefficient must be between 1 and 10. If it's not, you’ll need to adjust it by moving the decimal point and changing the exponent accordingly. Moving the decimal to the left increases the exponent, and moving it to the right decreases the exponent. Remember that balancing act we talked about? Keeping these steps in mind will make multiplying numbers in scientific notation a piece of cake. You've got this! Now that we’ve reviewed the basics, let's address some common questions you might have about scientific notation and multiplication.

Common Questions and Pitfalls

Okay, guys, let's talk about some common questions and pitfalls that often come up when dealing with scientific notation, especially when multiplying. One frequent question is,