Simplifying Expressions A Step-by-Step Guide To (2xy³)^¹ (6x²y)² / (2xy) (16x³y²)

Hey guys, let's dive into this math problem together! We've got a real doozy here: simplifying the expression (2xy³)^1 (6x²y)² / (2xy) (16x³y²). Don't worry, it looks intimidating, but we'll break it down step by step and make it super easy to understand. Think of it like untangling a messy ball of yarn – we'll just take it one piece at a time.

Breaking Down the Expression

First, let's focus on understanding the key concepts involved. We're dealing with exponents, variables, and fractions. Remember the basic rules of exponents: (a^m)n = a^(m*n) and a^m * a^n = a^(m+n). These will be our best friends in this journey. Also, when dividing terms with the same base, we subtract the exponents: a^m / a^n = a^(m-n). It's like magic, but it's just math!

Now, let’s rewrite the expression to make it clearer. We have (2xy³)^1 (6x²y)² in the numerator and (2xy) (16x³y²) in the denominator. This means we're multiplying terms in the numerator and then dividing the result by the product of terms in the denominator. It's all about following the order of operations, which, as you know, is super important in math. Think of it as the recipe for this mathematical dish we're cooking up – we need to follow the steps in the right order to get the delicious result!

Let's start by simplifying the numerator. The term (2xy³)^1 is pretty straightforward – it's just 2xy³. The exponent 1 doesn't change anything. However, (6x²y)² is where things get a bit more interesting. We need to apply the exponent to everything inside the parentheses. That means 6², (x²)², and y². So, (6x²y)² becomes 36x⁴y². Remember, when we raise a power to another power, we multiply the exponents. It's like leveling up the power!

Now, let's simplify the denominator. We have (2xy) (16x³y²). This is simpler – we just multiply the coefficients (the numbers) and add the exponents of the same variables. So, 2 * 16 = 32, x¹ * x³ = x⁴, and y¹ * y² = y³. Thus, the denominator becomes 32x⁴y³. We're halfway there, guys! We've tamed both the numerator and the denominator. Now comes the fun part – the grand finale of simplification!

Step-by-Step Simplification

Okay, let's get our hands dirty and simplify this beast of an expression. We start with the original expression:

(2xy³)^1 (6x²y)² / (2xy) (16x³y²)

First, we simplify each term individually. As we discussed, (2xy³)^1 is simply 2xy³. For (6x²y)², we square each part: 6² = 36, (x²)² = x⁴, and y² = y². So, (6x²y)² becomes 36x⁴y². In the denominator, we have (2xy) (16x³y²). Multiplying these gives us 2 * 16 = 32, x * x³ = x⁴, and y * y² = y³. So, the denominator simplifies to 32x⁴y³.

Now our expression looks like this:

(2xy³ * 36x⁴y²) / (32x⁴y³)

Next, we multiply the terms in the numerator. We multiply the coefficients: 2 * 36 = 72. For the variables, we add the exponents of the same base: x¹ * x⁴ = x⁵ and y³ * y² = y⁵. So, the numerator becomes 72x⁵y⁵.

Our expression is now:

72x⁵y⁵ / 32x⁴y³

Now comes the division part. We divide the coefficients: 72 / 32. Both 72 and 32 are divisible by 8, so we simplify the fraction to 9/4. For the variables, we subtract the exponents: x⁵ / x⁴ = x^(5-4) = x¹ = x and y⁵ / y³ = y^(5-3) = y². So, the simplified variable part is xy².

Putting it all together, we get:

(9/4)xy²

And that’s it! We've simplified the expression. It’s like we’ve taken a tangled mess and turned it into a neat, organized little package. Remember, the key is to break the problem down into smaller, manageable steps. Don't try to do everything at once – focus on one step at a time, and you'll get there.

Common Mistakes to Avoid

Hey, we all make mistakes, especially in math! But it's important to learn from them. One common mistake people make when simplifying expressions like this is forgetting to apply the exponent to all parts of a term. For example, when simplifying (6x²y)², some might forget to square the 6 and only focus on the variables. Remember, the exponent applies to everything inside the parentheses!

Another mistake is not following the correct order of operations. Remember PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction)? It's our guide through the mathematical wilderness. Make sure you simplify exponents before you multiply or divide. It’s like following the recipe in order – if you add the ingredients at the wrong time, the dish won’t turn out right!

Also, when dividing terms with the same base, make sure you subtract the exponents correctly. It’s easy to get mixed up and add them instead, but remember the rule: a^m / a^n = a^(m-n). Keep those rules handy, and you'll be golden.

Finally, always double-check your work! It's so easy to make a small mistake, like a sign error or a forgotten exponent. Taking a few extra seconds to review your steps can save you a lot of grief in the long run. Think of it as proofreading your work – you want to make sure everything is just right.

Practice Problems

Alright, guys, now that we've conquered this problem together, it's time for you to test your skills! Practice makes perfect, as they say. The more you work with these kinds of problems, the more comfortable you'll become with them. It's like learning a new language – the more you practice, the more fluent you become.

Here are a few practice problems for you to try:

1. Simplify: (3a²b)³ (2ab²) / (6a⁴b⁴)
2. Simplify: (5x³y²)² / (10x²y) (x⁴y³)
3. Simplify: (4mn²) (9m²n) / (12m³n³)

Work through these problems step by step, and remember the tips and tricks we discussed. If you get stuck, don't be afraid to go back and review the steps we took in the original problem. The key is to be patient and persistent. Math can be challenging, but it's also incredibly rewarding when you finally crack a tough problem.

Real-World Applications

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