Simplify (x^6 ⋅ Y^3 ⋅ X^3)^2 Equivalent Expression Explained

Hey guys! Today, we're diving into the fascinating world of algebraic expressions, and we're going to break down a problem that might look intimidating at first glance. Our mission is to find an expression that's equivalent to the given expression: (x^6 ⋅ y^3 ⋅ x³⁾2. Don't worry; we'll take it step by step, making sure everyone understands the process. We'll explore the fundamental rules of exponents and how to apply them to simplify this expression. By the end of this article, you'll not only know the answer but also understand the why behind it. So, let's get started and unlock the secrets of equivalent expressions!

Breaking Down the Expression: (x^6 ⋅ y^3 ⋅ x³⁾2

Let's start by really digging into the expression we've got: (x^6 ⋅ y^3 ⋅ x³⁾2. The first thing we want to do is simplify what's inside the parentheses. Think of it like cleaning up your room before you start rearranging the furniture. We have terms with 'x' and a term with 'y'. The key here is to remember the product of powers rule, which states that when you multiply powers with the same base, you add the exponents. In simpler terms, x^m * x^n = x^(m+n). So, let's apply this rule to our 'x' terms.

We have x^6 and x^3 inside the parentheses. When we multiply these, we add their exponents: 6 + 3 = 9. So, x^6 * x^3 becomes x^9. Now, our expression inside the parentheses looks like this: x^9 ⋅ y^3. See? We're already making progress! We've combined the 'x' terms, and now we just have x^9 and y^3 hanging out inside those parentheses. Next, we need to deal with that exponent outside the parentheses – the squared part. Remember, the entire expression inside the parentheses is being raised to the power of 2. This means we're going to use another important exponent rule: the power of a product rule. This rule says that (ab)^n = a^n * b^n. Basically, if you have a product raised to a power, you raise each factor in the product to that power. Applying this to our expression, we need to raise both x^9 and y^3 to the power of 2. This is where the power of a power rule comes into play. This rule states that (a^m)n = a^(m*n). When you raise a power to another power, you multiply the exponents. So, let's apply this to x^9 and y^3.

For x^9 raised to the power of 2, we multiply the exponents: 9 * 2 = 18. Therefore, (x⁹⁾2 becomes x^18. Similarly, for y^3 raised to the power of 2, we multiply the exponents: 3 * 2 = 6. So, (y³⁾2 becomes y^6. Now, we can put it all together. We started with (x^6 ⋅ y^3 ⋅ x³⁾2, simplified the inside to get x^9 ⋅ y^3, and then applied the outer exponent of 2. This gave us x^18 and y^6. Therefore, the equivalent expression is x^18 * y^6. We've successfully navigated the world of exponents and simplified our expression! Understanding these rules is super important for tackling more complex algebra problems. So, make sure you've got these exponent rules locked in your memory bank. You'll be using them a lot!

Exploring the Exponent Rules Used

Alright, let's dive deeper into the exponent rules that we used to simplify the expression (x^6 ⋅ y^3 ⋅ x³⁾2. Understanding these rules is like having a superpower in algebra – they allow you to manipulate and simplify complex expressions with ease. We're going to break down each rule, explain why it works, and show how we applied it in our problem. Think of this section as your personal guide to mastering exponents. First up, we have the Product of Powers Rule. This rule is the foundation for simplifying expressions when you're multiplying terms with the same base. It states that when you multiply powers with the same base, you add the exponents. Mathematically, it's expressed as x^m * x^n = x^(m+n). But why does this work? Let's think about what exponents actually mean. x^m means 'x' multiplied by itself 'm' times, and x^n means 'x' multiplied by itself 'n' times. So, when you multiply x^m and x^n, you're essentially multiplying 'x' by itself a total of 'm + n' times. That's why we add the exponents!

In our problem, we used this rule to combine x^6 and x^3 inside the parentheses. We had x^6 ⋅ x^3, and we added the exponents 6 and 3 to get x^(6+3), which simplified to x^9. See how the rule directly applies to our situation? It's all about understanding the underlying principle. Next, we encountered the Power of a Product Rule. This rule comes into play when you have a product raised to a power. It states that you can raise each factor in the product to that power individually. In mathematical terms, (ab)^n = a^n * b^n. Why is this the case? Well, (ab)^n means (ab) multiplied by itself 'n' times: (ab) * (ab) * (ab) ... (n times). You can rearrange these factors because multiplication is commutative (the order doesn't matter). So, you can group all the 'a's together and all the 'b's together: (a * a * a ... n times) * (b * b * b ... n times). This is the same as a^n * b^n. In our problem, this rule helped us deal with the outer exponent of 2. After simplifying the inside of the parentheses to x^9 ⋅ y^3, we had (x^9 ⋅ y³⁾2. The Power of a Product Rule tells us that we can apply the exponent 2 to both x^9 and y^3 separately. This led us to the next important rule: the Power of a Power Rule.

The Power of a Power Rule is crucial when you have a power raised to another power. It tells us that we multiply the exponents. Mathematically, this is represented as (a^m)n = a^(mn). Let's break down why this rule holds true. (a^m)n means a^m multiplied by itself 'n' times: (a^m) * (a^m) * (a^m) ... (n times). Remember, a^m means 'a' multiplied by itself 'm' times. So, we have 'a' multiplied by itself 'm' times, and we're doing this 'n' times. In total, 'a' is multiplied by itself mn times, which gives us a^(mn). In our problem, we used this rule to simplify (x⁹⁾2 and (y³⁾2. For (x⁹⁾2, we multiplied the exponents 9 and 2 to get x^(92), which simplifies to x^18. Similarly, for (y³⁾2, we multiplied the exponents 3 and 2 to get y^(3*2), which simplifies to y^6. By mastering these exponent rules – Product of Powers, Power of a Product, and Power of a Power – you'll be well-equipped to tackle a wide range of algebraic expressions. These rules are not just formulas to memorize; they're tools that help you understand the fundamental relationships between exponents and multiplication. Practice applying these rules in different scenarios, and you'll become a true exponent master!

Step-by-Step Solution to Find the Equivalent Expression

Okay, let's walk through the solution step-by-step, so it's crystal clear how we arrived at the answer. We're starting with the expression (x^6 ⋅ y^3 ⋅ x³⁾2. Remember, the key to solving these kinds of problems is to break them down into smaller, manageable steps. We're going to simplify the expression inside the parentheses first, then deal with the outer exponent. This approach makes the whole process much less daunting. Our first step is to focus on what's inside the parentheses: x^6 ⋅ y^3 ⋅ x^3. We notice that we have two terms with the same base, 'x'. This is where the Product of Powers Rule comes in handy. As we discussed earlier, this rule tells us that when we multiply powers with the same base, we add the exponents. So, we're going to combine x^6 and x^3. We add the exponents 6 and 3, which gives us 6 + 3 = 9. Therefore, x^6 ⋅ x^3 simplifies to x^9.

Now, let's rewrite the expression inside the parentheses with this simplification. We have x^9 ⋅ y^3. Notice that we can't simplify this any further because the bases are different ('x' and 'y'). So, we've done all we can with the inside of the parentheses. Our expression now looks like this: (x^9 ⋅ y³⁾2. The next step is to deal with the exponent outside the parentheses, the squared part. This means we need to raise the entire expression inside the parentheses to the power of 2. This is where the Power of a Product Rule and the Power of a Power Rule come into play. Remember, the Power of a Product Rule states that (ab)^n = a^n * b^n. This means we need to raise both x^9 and y^3 to the power of 2. So, we have (x⁹⁾2 and (y³⁾2. Now, we apply the Power of a Power Rule, which tells us that (a^m)n = a^(m*n). In other words, when we raise a power to another power, we multiply the exponents. Let's start with (x⁹⁾2. We multiply the exponents 9 and 2, which gives us 9 * 2 = 18. Therefore, (x⁹⁾2 simplifies to x^18.

Next, we tackle (y³⁾2. We multiply the exponents 3 and 2, which gives us 3 * 2 = 6. So, (y³⁾2 simplifies to y^6. Now, we can put it all together. We started with (x^6 ⋅ y^3 ⋅ x³⁾2, simplified the inside to get x^9 ⋅ y^3, and then applied the outer exponent of 2. This gave us x^18 and y^6. So, the equivalent expression is x^18 ⋅ y^6. To recap, we followed these steps: 1. Simplified inside the parentheses using the Product of Powers Rule. 2. Applied the outer exponent using the Power of a Product Rule and the Power of a Power Rule. By breaking the problem down into these steps, we made the simplification process much clearer and easier to follow. Remember, practice makes perfect! The more you work with these exponent rules, the more comfortable you'll become with applying them. So, don't be afraid to tackle similar problems and build your algebra skills.

Analyzing the Given Options

Now that we've simplified the expression (x^6 ⋅ y^3 ⋅ x³⁾2 to x^18 ⋅ y^6, let's take a look at the options provided and see which one matches our simplified expression. This is a crucial step in solving multiple-choice problems – you always want to compare your answer with the given choices to ensure you've arrived at the correct solution. Remember, the options were:

A. x|y| B. x{y} C. x^{25} y^6 D. (x y)^{\frac{17}{23}}

Let's analyze each option one by one. Option A, x|y|, involves the absolute value of 'y'. Absolute value means the distance of a number from zero, so it's always non-negative. This expression doesn't match our simplified expression x^18 ⋅ y^6, which involves exponents and doesn't have any absolute value. Therefore, option A is not the correct equivalent expression. Option B, x{y}, this appears to be a typo or an unconventional notation. In standard algebraic notation, this doesn't represent any valid mathematical operation. It's definitely not equivalent to x^18 ⋅ y^6, which involves powers of 'x' and 'y'. So, we can confidently eliminate option B.

Option C, x^{25} y^6, has 'x' raised to the power of 25 and 'y' raised to the power of 6. While the 'y^6' part matches our simplified expression, the 'x^{25}' part does not match our 'x^18'. Remember, we carefully simplified the expression, and we know that the exponent of 'x' should be 18. Therefore, option C is not the equivalent expression. Option D, (x y)^{\frac{17}{23}}, involves a fractional exponent and a product of 'x' and 'y' raised to that exponent. This form is quite different from our simplified expression x^18 ⋅ y^6. To see why, we can think about what a fractional exponent means. A fractional exponent like \frac{17}{23} represents both a root and a power. In this case, it would involve taking the 23rd root and then raising the result to the 17th power. This is a completely different operation than raising 'x' and 'y' to separate integer powers. Therefore, option D is not equivalent to our simplified expression. After analyzing all the options, we realize that there seems to be a mistake in the provided options. None of the options match our simplified expression, x^18 ⋅ y^6. This highlights an important point: it's always crucial to double-check your work and the given options. In a real-world scenario, if you encountered this situation, you would need to communicate the discrepancy. However, for the purpose of this exercise, we have correctly simplified the expression, and the equivalent expression is x^18 ⋅ y^6.

Common Mistakes to Avoid

Let's talk about some common mistakes people make when simplifying expressions like (x^6 ⋅ y^3 ⋅ x³⁾2. Knowing these pitfalls can help you avoid them and ensure you're on the right track. Think of this as a