Simplifying Algebraic Expressions: A Step-by-Step Guide

Hey guys! Today, we're diving into the world of algebraic expressions and tackling a common challenge: simplification. We're going to break down the process of simplifying the expression $(-2x^3)^2 \cdot y \cdot y^9$ step by step. This is a fundamental skill in mathematics, so let's get started and make sure you've got a solid grasp of it!

Understanding the Expression

Before we jump into simplifying, let's take a closer look at the expression we're dealing with: $(-2x^3)^2 \cdot y \cdot y^9$. It might look a bit intimidating at first, but don't worry! We can break it down into smaller, more manageable parts. The key here is to understand the order of operations and the rules of exponents. We've got parentheses, exponents, multiplication, and variables all mixed together, so it's crucial to know how each of these components works.

The first part of the expression, $(-2x^3)^2$, involves raising a term to a power. Remember that when you raise a product to a power, you need to apply the power to each factor within the parentheses. This means we'll be dealing with both the coefficient (-2) and the variable term ($x^3$). The second part involves multiplying variables with exponents: $y \cdot y^9$. When multiplying terms with the same base, we add the exponents. Knowing these basic rules is half the battle! So, let's keep these in mind as we go through the simplification process. We will expand on each of these rules as we apply them, ensuring a clear understanding. Simplifying algebraic expressions isn't just about getting the right answer; it's about building a solid foundation for more advanced math topics. So, let's take our time, understand each step, and you'll be simplifying like a pro in no time!

Step 1: Simplifying $(-2x^3)^2$

The first hurdle in simplifying our expression is dealing with the term $(-2x^3)^2$. Remember the rule: when you have a power raised to another power, you multiply the exponents. Also, when a product inside parentheses is raised to a power, each factor inside gets raised to that power. So, let's break it down:

• $(-2x^3)^2$ means we're squaring both -2 and $x^3$.
• $(-2)^2$ is simply $(-2) \times (-2)$, which equals 4. Remember, a negative number multiplied by a negative number gives a positive result. This is a crucial rule to remember when dealing with exponents and negative signs.
• For $(x^3)^2$, we use the power of a power rule: $(a^m)^n = a^{m \times n}$. In this case, it becomes $x^{3 \times 2} = x^6$. So, we're multiplying the exponents 3 and 2 to get 6. This rule is fundamental in simplifying expressions with exponents, and mastering it will make your life much easier.

Putting it all together, $(-2x^3)^2$ simplifies to $4x^6$. See? It's not so scary when we break it down step by step. The key is to focus on one rule at a time and apply it carefully. Now, let's move on to the next part of the expression, and we'll see how this simplified term fits into the bigger picture. We're making great progress, guys! Keep up the awesome work, and remember, practice makes perfect. The more you work with these rules, the more natural they will become. So, let's keep going!

Step 2: Simplifying y ullet y^9

Now, let's tackle the next part of our expression: $y \cdot y^9$. This involves multiplying variables with exponents, and there's a handy rule for this: when multiplying terms with the same base, you add the exponents. It's like they're joining forces to become one bigger exponent! Let's see how this works in our case:

• Remember that if a variable doesn't have an exponent written, it's understood to have an exponent of 1. So, $y$ is the same as $y^1$. This is a common point of confusion, so make sure you remember this little trick. It will save you from making mistakes in the future.
• Now we have $y^1 \cdot y^9$. According to the rule, we add the exponents: $1 + 9 = 10$.
• Therefore, $y^1 \cdot y^9$ simplifies to $y^{10}$. It's as simple as that! By adding the exponents, we've combined these two terms into a single, simplified term. This rule is super useful and comes up all the time in algebra, so it's definitely one to remember.

So far, we've simplified $(-2x^3)^2$ to $4x^6$ and $y \cdot y^9$ to $y^{10}$. We're well on our way to simplifying the entire expression. Remember, the key is to take it step by step and apply the rules of exponents carefully. Now, let's put these pieces together and see how the final simplification looks. We're almost there, guys! Just a little bit more to go, and you'll have mastered this expression.

Step 3: Combining the Simplified Terms

We've done the heavy lifting by simplifying the individual parts of the expression. Now comes the satisfying part: putting it all together! We found that $(-2x^3)^2$ simplifies to $4x^6$, and $y \cdot y^9$ simplifies to $y^{10}$. Our original expression was $(-2x^3)^2 \cdot y \cdot y^9$, so let's substitute our simplified terms back in:

• We replace $(-2x^3)^2$ with $4x^6$, and $y \cdot y^9$ with $y^{10}$.
• This gives us $4x^6 \cdot y^{10}$.

And that's it! We've simplified the expression. There are no more like terms to combine, and we've applied all the necessary exponent rules. The final simplified form of $(-2x^3)^2 \cdot y \cdot y^9$ is $4x^6y^{10}$. Guys, isn't it cool how a seemingly complex expression can be simplified down to something so neat and tidy? This is the power of algebra! It's like solving a puzzle, where each step brings you closer to the final solution. And the best part is, you've now got the skills to tackle similar problems. Remember, practice is key. The more you work with these concepts, the more confident you'll become. So, keep practicing, keep exploring, and you'll be amazed at what you can achieve!

Final Answer

So, to recap, we started with the expression $(-2x^3)^2 \cdot y \cdot y^9$ and, through a series of steps, simplified it to $4x^6y^{10}$. Here's a quick rundown of the steps we took:

1. Simplified $(-2x^3)^2$ to $4x^6$ by applying the power of a product rule and the power of a power rule.
2. Simplified $y \cdot y^9$ to $y^{10}$ by adding the exponents of like bases.
3. Combined the simplified terms to get the final answer: $4x^6y^{10}$.

Remember, the key to simplifying algebraic expressions is to break them down into smaller parts, apply the rules of exponents and operations carefully, and combine like terms. And most importantly, don't be afraid to make mistakes! Mistakes are part of the learning process. The more you practice and learn from your mistakes, the better you'll become at simplifying expressions. You've done a fantastic job following along with this example. Now, go out there and try some more problems on your own. You've got this! And who knows, maybe you'll even start to enjoy the challenge of simplifying algebraic expressions. Keep up the great work, and remember, math can be fun!