Simplify (-4)^3 * (-4)^2: Product Rule Explained

Hey guys! Ever wondered how to simplify those tricky expressions with exponents? Today, we're diving deep into the product rule of exponents. Specifically, we'll break down how to simplify the expression $(-4)^3 imes (-4)^2$. It might seem daunting at first, but trust me, with a clear understanding of the rules, it's a piece of cake! We will go through step by step explanation to make sure everyone understands every aspect of this rule. So, let's get started and make exponents our friends!

First things first, what exactly is the product rule of exponents? Simply put, the product rule of exponents states that when you're multiplying two powers with the same base, you add the exponents. Mathematically, it's expressed as: $a^m imes a^n = a^{m+n}$. This rule is fundamental in simplifying expressions and solving equations involving exponents. Think of it like this: exponents are just a shorthand way of writing repeated multiplication. So, when you multiply two expressions with the same base, you're essentially combining those repeated multiplications. Understanding this core concept makes it much easier to apply the rule correctly and avoid common mistakes. Now, let's take a closer look at how this rule works with our example expression and how we can apply it step by step. Remember, a solid grasp of the basics is key to tackling more complex problems later on.

Okay, let's tackle our expression: $(-4)^3 imes (-4)^2$. To simplify this using the product rule, we need to identify the base and the exponents. In this case, our base is -4, and our exponents are 3 and 2. According to the product rule, we should add the exponents together since the bases are the same. So, we have $(-4)^3 imes (-4)^2 = (-4)^{3+2}$. Now, let's do the simple addition: 3 + 2 = 5. This gives us $(-4)^5$. It's crucial to keep the base, which is -4, the same throughout the process. A common mistake is to change the base or misapply the addition. Remember, the product rule is all about adding the exponents when the bases are the same. So, the simplified form of our expression, using the product rule, is $(-4)^5$. We've taken it step by step, making sure each part is clear. Next, we'll take a look at what this actually means in terms of repeated multiplication and why the negative sign is so important.

Let's walk through the simplification step by step to ensure we understand every part of the process. We start with the expression $(-4)^3 imes (-4)^2$. The first thing to recognize is that both terms have the same base, which is -4. This is crucial because the product rule of exponents only applies when the bases are the same. Now, we apply the product rule, which tells us to add the exponents: $(-4)^3 imes (-4)^2 = (-4)^{3+2}$. Next, we perform the addition in the exponent: 3 + 2 = 5. So our expression becomes $(-4)^5$. This is the simplified form using the product rule. But what does this mean? $(-4)^5$ means -4 multiplied by itself five times: $(-4) imes (-4) imes (-4) imes (-4) imes (-4)$. If we calculate this, we get -1024. However, the question asks us to simplify using the product rule, so $(-4)^5$ is the correct simplified form. Remember, each step is important: identifying the base, applying the rule, and simplifying the exponent. Let's move on to discussing why the negative sign is so important in this context.

The negative sign in our base (-4) plays a crucial role in the final result. It's important to understand how negative numbers behave when raised to different powers. When a negative number is raised to an odd power, the result is negative. For example, $(-4)^3 = (-4) imes (-4) imes (-4) = -64$. The product of three negative numbers is negative. On the other hand, when a negative number is raised to an even power, the result is positive. For example, $(-4)^2 = (-4) imes (-4) = 16$. The product of two negative numbers is positive. In our case, we have $(-4)^5$. Since 5 is an odd number, the result will be negative. This is why the answer $(-4)^5$ is correct and $4^5$ would be incorrect. The negative sign must be included to accurately reflect the value of the expression. Always pay close attention to the sign of the base, as it significantly affects the outcome. Now, let’s compare this with some incorrect options to see why they don't work.

Let's analyze why the other options are incorrect to solidify our understanding. Option B, $(-4)^6$, is incorrect because we added the exponents 3 and 2, which equals 5, not 6. The correct exponent should be 5, not 6. Option C, $4^5$, is incorrect because it neglects the negative sign of the base. As we discussed, the negative sign is crucial when dealing with odd powers. $(-4)^5$ is not the same as $4^5$. Option D, $4^6$, is incorrect for both reasons: it neglects the negative sign and incorrectly adds the exponents. The correct simplification must include the negative sign and have the correct exponent. Understanding why these options are wrong helps reinforce the correct application of the product rule and the importance of the negative sign. It’s not just about getting the right answer; it’s about understanding the underlying principles. Now, let’s wrap things up with a quick recap and some final thoughts.

So, guys, we've successfully simplified the expression $(-4)^3 imes (-4)^2$ using the product rule of exponents. The correct answer is $(-4)^5$. We walked through the process step by step, emphasizing the importance of identifying the base, adding the exponents correctly, and paying close attention to the negative sign. Remember, the product rule states that when multiplying powers with the same base, you add the exponents. The negative sign is crucial because it affects the sign of the final result when the exponent is odd. By understanding these key concepts, you can confidently tackle similar problems. Practice makes perfect, so keep applying these rules to various expressions. Mastering the product rule of exponents is a fundamental skill in algebra and will help you in more advanced mathematical topics. Keep practicing, and you’ll become an exponent expert in no time!