Step-by-Step Guide Solving (4² × (-5)²)³

Hey guys! Today, we're diving into a math problem that might look intimidating at first, but trust me, we'll break it down step by step until it feels like a breeze. We're going to tackle the expression (4² × (-5)²)³. So, grab your pencils, open your notebooks, and let's get started on this mathematical adventure!

Understanding the Order of Operations

Before we even think about plugging in numbers, it's crucial we understand the order of operations. This is the golden rule that dictates how we solve mathematical expressions, ensuring we all arrive at the same correct answer. Remember PEMDAS/BODMAS? It's our trusty acronym:

• Parentheses / Brackets
• Exponents / Orders
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

This order isn't just a suggestion; it's the law in the math world. Ignoring it can lead to some seriously wrong answers, and we definitely don't want that. So, keep PEMDAS/BODMAS in the back of your mind as we move forward. With a solid grasp on the order of operations, we're setting ourselves up for success in solving this problem. Understanding this principle ensures that we approach the problem systematically, breaking it down into manageable parts. Now, let’s see how it applies to our specific expression, making the whole process much clearer and less daunting.

Breaking Down the Expression

Okay, let's take a closer look at our expression: (4² × (-5)²)³. We've got parentheses, exponents, and multiplication all bundled together, and then the whole thing is raised to another power! But don't worry; PEMDAS/BODMAS is our guide. The first thing we need to do according to PEMDAS is tackle what's inside the parentheses. Inside the parentheses, we see two terms with exponents: 4² and (-5)². This is where our understanding of exponents comes into play. Remember, an exponent tells us how many times to multiply a number by itself. So, 4² means 4 multiplied by itself, and (-5)² means -5 multiplied by itself. Once we've dealt with the exponents inside the parentheses, we'll move on to the multiplication. We'll multiply the results of 4² and (-5)² together. This will simplify the expression inside the parentheses to a single number. Finally, after handling the parentheses and the operations within, we'll deal with the exponent outside the parentheses. This means we'll take the single number we got from simplifying the inside and raise it to the power of 3. This last step will give us our final answer. By breaking the problem down in this way, following PEMDAS/BODMAS, we make sure we handle each operation in the correct sequence, paving the way for an accurate solution.

Step 1: Evaluating the Exponents Inside the Parentheses

The first task at hand is to simplify the terms inside the parentheses, and that means dealing with the exponents. We have 4² and (-5)². Let's tackle these one at a time. Remember, an exponent tells us how many times to multiply the base by itself. So, 4² simply means 4 multiplied by 4. Mathematically, we write this as 4 * 4. And what does that equal? It's 16, of course! So, we've successfully evaluated 4² and found it to be 16. Now, let's move on to the next term inside the parentheses: (-5)². This means -5 multiplied by -5. When we multiply two negative numbers together, the result is a positive number. So, -5 multiplied by -5 equals positive 25. Therefore, (-5)² evaluates to 25. We've now simplified both exponential terms inside the parentheses. We've found that 4² is 16 and (-5)² is 25. This is a crucial step because it reduces the complexity of the expression, making it easier to manage in the subsequent steps. By correctly evaluating these exponents, we're one step closer to solving the entire problem. Next, we'll use these simplified values to perform the multiplication within the parentheses, further streamlining our expression and bringing us closer to the final answer.

Step 2: Performing the Multiplication Inside the Parentheses

Alright, we've successfully evaluated the exponents inside the parentheses. We know that 4² = 16 and (-5)² = 25. Now, it's time to move on to the next operation within the parentheses: multiplication. Our expression inside the parentheses is now 16 × 25. This is a straightforward multiplication problem, but it's important to get it right to ensure our final answer is correct. We need to multiply 16 by 25. You can do this manually, use a calculator, or even break it down into smaller multiplications if that's easier for you. For example, you could multiply 16 by 20 and then 16 by 5, and add the results. However you choose to do it, the result of 16 multiplied by 25 is 400. So, the expression inside the parentheses simplifies to 400. This is a significant step forward. We've taken the original, more complex expression inside the parentheses and condensed it down to a single number. By performing this multiplication accurately, we've simplified the problem considerably and are now in a much better position to tackle the final exponent. This step highlights the power of following the order of operations. By addressing the multiplication inside the parentheses before dealing with the outer exponent, we've made the problem much more manageable.

Step 3: Evaluating the Outer Exponent

Okay, we've made excellent progress! We've simplified the expression inside the parentheses down to a single number: 400. Now, it's time to deal with the final boss: the exponent outside the parentheses. Our expression now looks like this: (400)³. This means we need to raise 400 to the power of 3. Remember, an exponent tells us how many times to multiply the base by itself. So, 400³ means 400 multiplied by itself three times. Mathematically, we can write this as 400 * 400 * 400. Now, this is a larger multiplication, but we can still handle it. Let's break it down a bit. First, let's multiply 400 by 400. This gives us 160,000. Now, we need to multiply that result by 400 again. So, we have 160,000 * 400. This multiplication gives us a final result of 64,000,000. Therefore, (400)³ evaluates to 64,000,000. We've done it! We've successfully evaluated the outer exponent and arrived at our final answer. This step was crucial because it involved dealing with a large number and a higher power, but by breaking it down into smaller multiplications, we were able to tackle it effectively. This demonstrates the importance of understanding exponents and how they work, as well as the power of breaking down complex problems into simpler steps.

Final Answer: 64,000,000

And there you have it, guys! After carefully following the order of operations and breaking down the problem step by step, we've arrived at the solution. The expression (4² × (-5)²)³ evaluates to a whopping 64,000,000. That's sixty-four million! It might seem like a huge number, but we conquered it together. Remember, the key to solving complex mathematical problems is to take them one step at a time. By understanding the order of operations (PEMDAS/BODMAS) and breaking the problem down into manageable chunks, we can tackle even the most intimidating expressions. We started by simplifying the exponents inside the parentheses, then performed the multiplication, and finally, we dealt with the exponent outside the parentheses. Each step built upon the previous one, leading us to our final answer. This problem is a great example of how mathematics can be challenging but also rewarding. By sticking with it and applying the correct principles, we were able to find the solution and gain a deeper understanding of mathematical operations. So, the next time you encounter a problem like this, remember our step-by-step approach, and you'll be well on your way to solving it!