Simplifying Algebraic Expressions A Step-by-Step Guide To (-3x³yz³)²(2x²y)⁴

Hey guys! Let's dive into this interesting mathematical expression: (-3x³yz³)²(2x²y)⁴. It looks a bit intimidating at first, but don't worry, we'll break it down step by step. Our goal here is not just to find the simplified form but also to truly understand the mechanics behind the simplification. We'll explore the fundamental rules of exponents and how they apply in this specific case. By the end of this article, you'll not only be able to solve this problem but also tackle similar algebraic expressions with confidence. This isn't just about crunching numbers; it's about developing a strong foundation in algebra. So, grab your thinking caps, and let's get started!

Breaking Down the Expression

Let's begin by understanding the components of our expression. We have two main parts: (-3x³yz³)² and (2x²y)⁴. Each of these parts involves a combination of numerical coefficients and variables raised to different powers. The key to simplifying this expression lies in applying the rules of exponents correctly. Remember, when we raise a product to a power, we raise each factor in the product to that power. This is a crucial rule that we'll use repeatedly. For instance, (ab)ⁿ = aⁿbⁿ. Also, when we raise a power to a power, we multiply the exponents: (aᵐ)ⁿ = aᵐⁿ. These are the building blocks we'll use to dissect this problem. Understanding these rules isn't just about memorization; it's about grasping the underlying principles of how exponents work. Think of it like this: exponents are shorthand for repeated multiplication, and these rules are simply ways to keep track of all those multiplications. As we delve deeper, we'll see how these rules come into play, making the seemingly complex expression much more manageable. Let's start by tackling the first part: (-3x³yz³)².

Simplifying (-3x³yz³)²

The first part of our expression is (-3x³yz³)². To simplify this, we need to apply the power of a product rule, which states that (ab)ⁿ = aⁿbⁿ. This means we'll raise each factor inside the parentheses to the power of 2. So, let's break it down: (-3)² * (x³)² * y² * (z³)²

• (-3)²: This means -3 multiplied by itself, which equals 9.
• (x³)²: Here, we apply the power of a power rule, which states that (aᵐ)ⁿ = aᵐⁿ. So, we multiply the exponents: 3 * 2 = 6. This gives us x⁶.
• y²: This term is already in its simplest form, so we just keep it as y².
• (z³)²: Again, we use the power of a power rule: 3 * 2 = 6. This gives us z⁶.

Now, let's put it all together: 9x⁶y²z⁶. So, (-3x³yz³)² simplifies to 9x⁶y²z⁶. We've successfully handled the first part! Notice how we meticulously applied each rule, ensuring we didn't miss any steps. This step-by-step approach is crucial in algebra, preventing errors and fostering a clear understanding. Remember, each term within the parentheses is treated individually, and the exponent outside affects each one. Now that we've conquered the first part, let's move on to the second part of our expression: (2x²y)⁴. We'll use the same principles here, but with a different exponent, so pay close attention to how the rules adapt.

Simplifying (2x²y)⁴

Now, let's tackle the second part of the expression: (2x²y)⁴. Just like before, we'll use the power of a product rule, (ab)ⁿ = aⁿbⁿ, to simplify this. We need to raise each factor inside the parentheses to the power of 4. So, let's break it down:

• 2⁴: This means 2 multiplied by itself four times: 2 * 2 * 2 * 2 = 16.
• (x²)⁴: Applying the power of a power rule, (aᵐ)ⁿ = aᵐⁿ, we multiply the exponents: 2 * 4 = 8. This gives us x⁸.
• y⁴: This term is already in its simplest form, so we keep it as y⁴.

Putting it all together, we get 16x⁸y⁴. So, (2x²y)⁴ simplifies to 16x⁸y⁴. See how the process is becoming more familiar? We're essentially repeating the same steps, but with different numbers. This is the beauty of algebra: once you understand the fundamental rules, you can apply them to a wide variety of problems. The key is to be methodical and consistent in your approach. Now that we've simplified both parts of our expression, we're ready for the final step: combining the simplified terms. This is where we'll see how the product of powers rule comes into play, and we'll finally arrive at our final simplified expression.

Combining the Simplified Expressions

We've simplified (-3x³yz³)² to 9x⁶y²z⁶ and (2x²y)⁴ to 16x⁸y⁴. Now, we need to combine these two simplified expressions by multiplying them together: (9x⁶y²z⁶)(16x⁸y⁴)

To do this, we'll multiply the coefficients and then multiply the variables with the same base. Remember the product of powers rule: aᵐ * aⁿ = aᵐ⁺ⁿ. This rule is crucial here, as it tells us that when multiplying variables with the same base, we add their exponents.

• Multiply the coefficients: 9 * 16 = 144
• Multiply the x terms: x⁶ * x⁸ = x⁶⁺⁸ = x¹⁴
• Multiply the y terms: y² * y⁴ = y²⁺⁴ = y⁶
• The z term: z⁶ remains as it is since there's no other z term to multiply with.

Now, let's put it all together: 144x¹⁴y⁶z⁶. And there you have it! We've successfully simplified the original expression. This final step highlights the importance of understanding the interplay between different exponent rules. We used the product of powers rule to combine terms, solidifying our grasp of how exponents behave in multiplication. This isn't just about getting the right answer; it's about building a cohesive understanding of algebraic principles. So, let's recap what we've done and highlight the key takeaways from this exercise.

Final Simplified Expression and Key Takeaways

The final simplified expression is 144x¹⁴y⁶z⁶. Wow, we made it! We started with a complex expression and, through careful application of exponent rules, arrived at a much simpler form. This journey highlights several key takeaways:

• Understanding the Power of a Product Rule: (ab)ⁿ = aⁿbⁿ is fundamental. It allows us to distribute the exponent to each factor within the parentheses.
• Mastering the Power of a Power Rule: (aᵐ)ⁿ = aᵐⁿ is equally crucial. It helps us simplify expressions where a power is raised to another power.
• Applying the Product of Powers Rule: aᵐ * aⁿ = aᵐ⁺ⁿ is essential for combining terms with the same base.
• The Importance of a Step-by-Step Approach: Breaking down the problem into smaller, manageable steps makes the simplification process less daunting and reduces the chance of errors.
• Consistency is Key: Applying the rules consistently throughout the process ensures accuracy and builds confidence.

This exercise wasn't just about simplifying one expression; it was about reinforcing your understanding of core algebraic principles. These principles are the foundation upon which more advanced mathematical concepts are built. By mastering these basics, you're setting yourself up for success in future mathematical endeavors. So, keep practicing, keep exploring, and most importantly, keep asking questions! Math is a journey of discovery, and every problem solved is a step forward.