Simplifying Algebraic Expressions 2a^2b^3(4a^2 + 3ab^2 - Ab)

Hey everyone! Let's break down this math problem together. We've got a seemingly complex expression here: $2a^2b^3(4a^2 + 3ab^2 - ab)$. Don't worry; we'll simplify it step-by-step. Our goal is to distribute the term outside the parenthesis to each term inside, and then combine like terms. This is a classic application of the distributive property, and we'll make sure to handle the exponents correctly along the way. So, grab your pencils, and let's get started!

Understanding the Distributive Property

Before we dive into the problem, let's quickly review the distributive property. This fundamental concept in algebra states that $a(b + c) = ab + ac$. Essentially, it means we multiply the term outside the parentheses by each term inside. This principle extends to expressions with multiple terms, like the one we're dealing with. In our case, we'll be multiplying $2a^2b^3$ by each of the three terms inside the parentheses: $4a^2$, $3ab^2$, and $-ab$.

The distributive property is a cornerstone of algebraic manipulation, allowing us to expand and simplify expressions. It's not just about multiplying numbers; it's also about handling variables and their exponents correctly. When multiplying terms with exponents, remember the rule: $x^m * x^n = x^(m+n)$. This means we add the exponents of like variables. For instance, $a^2 * a^2$ becomes $a^(2+2) = a^4$. Keeping these rules in mind will help us avoid common mistakes and ensure we arrive at the correct simplified expression. Now that we've refreshed our understanding of the distributive property, we're ready to tackle the problem head-on!

Step-by-Step Simplification

Okay, let's get our hands dirty and simplify the expression $2a^2b^3(4a^2 + 3ab^2 - ab)$. We'll take it one term at a time, making sure we're clear on each step.

1. First Term: Multiply $2a^2b^3$ by $4a^2$. Remember to multiply the coefficients (the numbers in front) and add the exponents of like variables:

   • $2 * 4 = 8$
   • $a^2 * a^2 = a^(2+2) = a^4$
   • $b^3$ remains as is since there's no 'b' term in $4a^2$
   • So, the first term becomes $8a^4b^3$

2. Second Term: Multiply $2a^2b^3$ by $3ab^2$:

   • $2 * 3 = 6$
   • $a^2 * a = a^(2+1) = a^3$
   • $b^3 * b^2 = b^(3+2) = b^5$
   • The second term is $6a^3b^5$

3. Third Term: Multiply $2a^2b^3$ by $-ab$. Don't forget the negative sign!

   • $2 * -1 = -2$
   • $a^2 * a = a^(2+1) = a^3$
   • $b^3 * b = b^(3+1) = b^4$
   • The third term is $-2a^3b^4$

Now, we combine all these terms together. This process ensures that each part of the original expression is correctly accounted for in the simplified version. By breaking down the multiplication into smaller, manageable steps, we minimize the chances of errors and maintain clarity throughout the simplification process.

Combining the Terms

Alright, we've multiplied $2a^2b^3$ by each term inside the parentheses. Now, let's put it all together. We found that:

• $2a^2b^3 * 4a^2 = 8a^4b^3$
• $2a^2b^3 * 3ab^2 = 6a^3b^5$
• $2a^2b^3 * -ab = -2a^3b^4$

So, when we combine these, we get:

$8a^4b^3 + 6a^3b^5 - 2a^3b^4$

Now, let's take a quick look to see if there are any like terms we can combine. Like terms have the same variables raised to the same powers. In this case, we have terms with $a^4b^3$, $a^3b^5$, and $a^3b^4$. Notice that the exponents for 'a' and 'b' are different in each term, which means they are not like terms. Therefore, we can't simplify this expression any further. We've successfully distributed and combined, and we're left with our simplified expression!

Identifying the Correct Answer

Okay, guys, we've simplified the expression $2a^2b^3(4a^2 + 3ab^2 - ab)$ to $8a^4b^3 + 6a^3b^5 - 2a^3b^4$. Now, let's match this with the answer choices provided:

A) $8a^4b^5 + 3a^3b^5 - 2a^3b^4$ B) $8a^4b^3 + 6a^3b^5 + 2a^3b^4$ C) $8a^4b^5 + 3a^3b^5 + 2a^3b^4$ D) $8a^4b^3 + 6a^3b^5 - 2a^3b^4$

By carefully comparing our result with the options, we can see that option D exactly matches our simplified expression: $8a^4b^3 + 6a^3b^5 - 2a^3b^4$. The key here is to pay close attention to the signs and exponents. A small difference can lead to an incorrect answer. So, always double-check your work and compare it meticulously with the given choices. We've found our correct answer!

Common Mistakes to Avoid

When simplifying expressions like this, there are a few common traps that students often fall into. Let's highlight these so you can steer clear of them!

1. Forgetting the Distributive Property: The most fundamental mistake is not distributing the term outside the parentheses to every term inside. Make sure you multiply $2a^2b^3$ by each of the three terms: $4a^2$, $3ab^2$, and $-ab$.
2. Incorrectly Handling Exponents: Remember the rule: when multiplying terms with the same base, you add the exponents. For example, $a^2 * a$ is $a^3$, not $a^2$. Similarly, $b^3 * b^2$ is $b^5$, not $b^6$. A common error is to multiply the exponents instead of adding them.
3. Sign Errors: Pay close attention to the signs, especially when dealing with negative terms. Multiplying by a negative number changes the sign of the term. For example, $2a^2b^3 * -ab = -2a^3b^4$. A simple sign error can throw off the entire calculation.
4. Combining Unlike Terms: Only combine terms that have the same variables raised to the same powers (like terms). For example, $8a^4b^3$ and $6a^3b^5$ cannot be combined because the exponents of 'a' and 'b' are different. Attempting to combine unlike terms leads to an incorrect simplification.

By being mindful of these common mistakes, you can increase your accuracy and confidence in simplifying algebraic expressions.

Practice Makes Perfect

The best way to master simplifying algebraic expressions is through practice. The more you work through problems, the more comfortable you'll become with the process and the less likely you are to make mistakes. So, here are a few tips for effective practice:

1. Start with Simple Problems: Begin with expressions that have fewer terms and lower exponents. This will help you build a solid foundation and understand the basic principles before moving on to more complex problems.
2. Work Through Examples: Look for worked examples in your textbook or online. Follow each step carefully and try to understand the reasoning behind it. Then, try solving similar problems on your own.
3. Vary the Types of Problems: Practice simplifying expressions with different numbers of terms, different exponents, and different combinations of variables. This will expose you to a wide range of scenarios and help you develop a versatile skill set.
4. Check Your Answers: Always check your answers to make sure you're on the right track. If you're unsure, ask a teacher, tutor, or classmate for help. It's better to correct mistakes early on than to reinforce incorrect methods.
5. Use Online Resources: There are many excellent online resources available, such as Khan Academy and other educational websites, that offer practice problems and step-by-step solutions. Take advantage of these resources to supplement your learning.

Remember, simplifying algebraic expressions is a skill that builds over time. With consistent practice and attention to detail, you'll become proficient at it. Keep up the hard work, and you'll see your skills improve!

Conclusion

So, there you have it! We successfully simplified the expression $2a^2b^3(4a^2 + 3ab^2 - ab)$ and found that the correct answer is D) $8a^4b^3 + 6a^3b^5 - 2a^3b^4$. We walked through the distributive property, carefully multiplied each term, combined like terms, and identified the answer. Remember to pay attention to exponents, signs, and the distributive property to avoid common mistakes. Keep practicing, and you'll become a pro at simplifying expressions! You got this!