Solving 4(-2.5)⁴ × (-2.5)³ A Step-by-Step Math Guide

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Hey guys! Today, we’re diving into a math problem that might look a little intimidating at first glance, but I promise, we’ll break it down step by step until it’s super clear. We’re going to tackle the expression 4(-2.5)⁴ × (-2.5)³. So, grab your thinking caps, and let's get started!

Understanding the Problem

Before we jump into solving, let's make sure we understand exactly what we're looking at. The expression 4(-2.5)⁴ × (-2.5)³ involves a few key components:

  • A constant: 4
  • A negative decimal base: -2.5
  • Exponents: ⁴ (to the power of 4) and ³ (to the power of 3)
  • Multiplication: The × symbol indicates we need to multiply these terms together.

When we see exponents, it's crucial to remember what they mean. An exponent tells us how many times to multiply the base by itself. For instance, (-2.5)⁴ means -2.5 multiplied by itself four times: (-2.5) × (-2.5) × (-2.5) × (-2.5). Similarly, (-2.5)³ means -2.5 multiplied by itself three times: (-2.5) × (-2.5) × (-2.5).

The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), will be our guide here. It dictates the sequence in which we perform calculations to ensure we arrive at the correct answer. According to PEMDAS, we need to deal with the exponents before we handle the multiplication. This means we'll first calculate (-2.5)⁴ and (-2.5)³ separately, and then we'll multiply the results by 4.

Now, why is understanding this initial setup so important? Well, math problems, especially those involving exponents and negative numbers, can become tricky if we don't have a clear strategy. Misunderstanding the exponents or forgetting the rules for multiplying negative numbers can lead to errors. By breaking down the expression into its components and understanding the order of operations, we set ourselves up for success. We're not just blindly crunching numbers; we're thinking through the problem logically.

Another key thing to note is the role of the negative sign. Since -2.5 is raised to both an even power (4) and an odd power (3), we need to be mindful of how the negative sign interacts with the exponents. A negative number raised to an even power becomes positive, while a negative number raised to an odd power remains negative. This is a fundamental rule that we'll apply as we solve the problem. By recognizing this early on, we avoid potential sign errors later. Alright, with the groundwork laid, we're ready to dive into the actual calculations. Let's move on to the next step where we'll tackle those exponents!

Step 1: Calculate (-2.5)⁴

Okay, let's start by figuring out what (-2.5)⁴ is. Remember, this means we're multiplying -2.5 by itself four times: (-2.5) × (-2.5) × (-2.5) × (-2.5). To make things easier, we can break this down into smaller steps.

First, let's multiply the first two terms: (-2.5) × (-2.5). When you multiply two negative numbers, you get a positive result. So, we have 2.5 × 2.5. If you do the math (or use a calculator), you'll find that 2.5 × 2.5 = 6.25. So, (-2.5) × (-2.5) = 6.25.

Now, we need to multiply the next two terms: (-2.5) × (-2.5), which we already know is 6.25. So, we have reduced our initial calculation to multiplying 6.25 by 6.25. This is because we have essentially paired the original four instances of -2.5 into two pairs, and then multiplied each pair. This strategy simplifies the process and reduces the chances of making mistakes. By breaking down the problem into manageable chunks, we can focus on accuracy at each step.

Next, we multiply 6.25 × 6.25. This is a bit more involved, but you can do it by hand or use a calculator. If you multiply it out, you'll find that 6.25 × 6.25 = 39.0625. Therefore, (-2.5)⁴ = 39.0625. Notice that the result is positive, which aligns with our earlier understanding that a negative number raised to an even power becomes positive. Double-checking such rules helps ensure we're on the right track.

So, we've successfully calculated the first exponent. Breaking the multiplication into smaller steps helped us keep track of the numbers and avoid errors. This approach is generally useful for larger exponents or more complex calculations. By handling the multiplication in stages, we can focus on the accuracy of each step before moving on to the next. Now that we know (-2.5)⁴ = 39.0625, we're halfway there! Let's move on to calculating the next exponent, (-2.5)³, in the next step. Remember, we're taking this one step at a time, making sure we understand each part of the problem before moving on. This is the key to tackling any math challenge!

Step 2: Calculate (-2.5)³

Alright, let's tackle the second exponent: (-2.5)³. This means we need to multiply -2.5 by itself three times: (-2.5) × (-2.5) × (-2.5). We've already done part of this calculation in the previous step, which will make things a bit easier.

From Step 1, we know that (-2.5) × (-2.5) = 6.25. So now, we just need to multiply this result by -2.5. This gives us 6.25 × (-2.5). Remember, when we multiply a positive number by a negative number, the result is negative.

Now, let's perform the multiplication: 6.25 × 2.5. You can do this by hand or use a calculator. If you multiply it out, you'll find that 6.25 × 2.5 = 15.625. But remember, since we're multiplying a positive by a negative, our result will be negative. Therefore, 6.25 × (-2.5) = -15.625.

So, (-2.5)³ = -15.625. Notice that this result is negative, which is what we expected since we're raising a negative number to an odd power. Keeping track of the signs is crucial in these types of calculations. A small sign error can throw off the entire result, so it's always a good idea to double-check that your answer makes sense in the context of the problem.

We've now successfully calculated (-2.5)³. By building on our previous calculation, we saved ourselves some time and effort. This is a common strategy in math: look for ways to reuse intermediate results to simplify the overall process. It not only makes the calculations faster but also reduces the chances of making errors. With both exponents calculated, we're now ready to move on to the final step, where we'll multiply everything together. Let's bring it all home and get to the solution!

Step 3: Multiply All the Terms Together

Okay, we're in the home stretch! We've calculated (-2.5)⁴ and (-2.5)³, and now it's time to put it all together. Our original expression was 4(-2.5)⁴ × (-2.5)³. We found that (-2.5)⁴ = 39.0625 and (-2.5)³ = -15.625. So, we can substitute these values back into the expression:

4 × 39.0625 × (-15.625)

To make things easier, let's first multiply 4 by 39.0625. If you do the math (or use a calculator), you'll find that 4 × 39.0625 = 156.25. So now our expression looks like this:

156.25 × (-15.625)

Now, we need to multiply 156.25 by -15.625. Again, remember that when we multiply a positive number by a negative number, the result is negative. This is a critical point to keep in mind to avoid sign errors. The multiplication itself is a bit hefty, so you'll likely want to use a calculator for this step.

If you multiply 156.25 by 15.625, you'll get 2441.40625. But since we're multiplying a positive number by a negative number, our final result will be negative. Therefore, 156.25 × (-15.625) = -2441.40625.

And there you have it! We've successfully multiplied all the terms together. So, the final answer to the expression 4(-2.5)⁴ × (-2.5)³ is -2441.40625. Woohoo! We made it through a fairly complex calculation by breaking it down into manageable steps and paying close attention to the order of operations and the rules for multiplying negative numbers.

Final Answer

So, after all that step-by-step calculation, we've arrived at our final answer. The solution to 4(-2.5)⁴ × (-2.5)³ is:

-2441.40625

Key takeaways:

  • Order of Operations (PEMDAS): Always follow the correct order of operations. Exponents before multiplication.
  • Negative Numbers: A negative number raised to an even power is positive; to an odd power, it’s negative.
  • Step-by-Step: Breaking the problem into smaller steps makes it much easier to manage and reduces errors.
  • Double-Check: Always double-check your calculations, especially the signs.

I hope this step-by-step explanation helped you understand how to solve this type of problem. Remember, math can seem daunting, but breaking it down and taking it one step at a time can make it much more approachable. Keep practicing, and you'll get the hang of it! You guys got this! If you have any questions, feel free to ask. Happy calculating!