Solving Scientific Notation Problems A Step By Step Guide

Hey guys! Let's dive into a common math problem: multiplying numbers in scientific notation. It might seem tricky at first, but I promise it's super manageable once you break it down. We're going to tackle this specific problem: (3.41 × 10⁻²) × (2.3 × 10⁻³). Not only will we solve it, but we'll also express the answer in scientific notation, just like pros! Ready to become scientific notation whizzes? Let's get started!

Understanding Scientific Notation

Before we jump into the calculation, let's quickly refresh what scientific notation actually means. Scientific notation is a neat way to write very large or very small numbers in a compact and standardized form. Think of it as a mathematical shorthand! It's expressed as the product of two parts:

• A coefficient: This is a number usually between 1 and 10 (it can be 1, but it has to be less than 10). It represents the significant digits of the number.
• A power of 10: This part indicates how many places the decimal point needs to be moved to get the original number. It's written as 10 raised to an exponent.

For example, the number 3,000,000 can be written in scientific notation as 3 × 10⁶. The coefficient is 3, and the power of 10 is 10⁶ (which means 10 to the power of 6). The exponent 6 tells us that the decimal point in 3 needs to be moved six places to the right to get 3,000,000. Similarly, a very small number like 0.0000025 can be written as 2.5 × 10⁻⁶. Notice the negative exponent! It indicates that the decimal point needs to be moved six places to the left.

Scientific notation is used extensively in various fields like science, engineering, and mathematics because it makes dealing with extremely large or small numbers much easier. Imagine trying to write the distance to a galaxy in standard form – you'd need a lot of zeros! Scientific notation helps us avoid that mess and express these numbers in a more concise and readable way. So, with this basic understanding, let's get back to our original problem and see how we can use scientific notation to solve it.

Step-by-Step Solution: (3.41 × 10⁻²) × (2.3 × 10⁻³)

Okay, guys, let's break down this problem step by step. Don't worry, it's easier than it looks! We're going to multiply (3.41 × 10⁻²) by (2.3 × 10⁻³). Here’s how we’ll do it:

1. Multiply the Coefficients

First, we're going to focus on the coefficients, which are the numbers in front of the powers of 10. In our problem, the coefficients are 3.41 and 2.3. So, let's multiply them:

3. 41 × 2.3 = 7.843

This is a straightforward multiplication, just like you’d do with any decimal numbers. You can use a calculator or do it by hand – whichever you prefer. The key thing is to get the product of these two coefficients. This result, 7.843, will be part of our final answer in scientific notation. Remember, the coefficient in scientific notation should be between 1 and 10, and 7.843 fits perfectly within this range.

2. Multiply the Powers of 10

Next, we'll deal with the powers of 10. We have 10⁻² and 10⁻³. Remember the rules of exponents? When you multiply powers with the same base (in this case, 10), you add the exponents. So, we have:

10. ⁻² × 10⁻³ = 10⁽⁻² ⁺ ⁻³⁾ = 10⁻⁵

This is a crucial step. Adding the exponents is what simplifies the powers of 10. Here, we added -2 and -3, which gave us -5. So, we now have 10⁻⁵ as our power of 10. This tells us that the decimal point will need to be moved five places to the left in the final answer.

3. Combine the Results

Now that we've multiplied the coefficients and the powers of 10 separately, let's put them together. We have:

7. 843 (from multiplying the coefficients) and 10⁻⁵ (from multiplying the powers of 10).

So, our result is:

7.843 × 10⁻⁵

And there you have it! We've successfully multiplied the two numbers in scientific notation. The final answer is 7.843 × 10⁻⁵, which is already in proper scientific notation form. The coefficient, 7.843, is between 1 and 10, and we have a clear power of 10. This completes our calculation, and we’ve expressed the result in the correct scientific notation format.

Final Answer in Scientific Notation

Alright, guys, we've done the hard work! We multiplied the coefficients, handled the powers of 10, and combined the results. So, what’s the final answer in scientific notation? Drumroll, please...

(3. 41 × 10⁻²) × (2.3 × 10⁻³) = 7.843 × 10⁻⁵

That's it! We've successfully calculated the product and expressed it in scientific notation. The coefficient is 7.843, and the exponent is -5. This means that the number is 0.00007843 in standard decimal form. But as you can see, scientific notation makes it much easier to write and understand this small number.

Key Takeaway: When multiplying numbers in scientific notation, multiply the coefficients and add the exponents of the powers of 10. Always make sure your final answer is in the correct scientific notation form, with the coefficient between 1 and 10.

Why Scientific Notation Matters

So, we've solved the problem, but you might be wondering,