Simplifying (2a³) × 3(a²+b²)³ × 5b⁴ A Step-by-Step Guide

Hey guys! Ever feel like you're staring at an algebraic expression that looks like it belongs in a math monster movie? Don't worry; we've all been there. Today, we're going to break down a seemingly complex problem into bite-sized pieces. We'll focus on simplifying the expression (2a³) × 3(a²+b²)³ × 5b⁴. Trust me, with a step-by-step approach, it’s totally manageable! We’ll not only solve this particular problem but also equip you with the tools to tackle similar algebraic challenges. So, let's dive in and make math a little less scary and a lot more fun!

Understanding the Basics

Before we jump into the main event, let's quickly recap some fundamental concepts. Remember, algebra is all about playing with symbols and numbers. Our goal is to make expressions simpler and easier to understand. When simplifying algebraic expressions, the key is to follow the order of operations (PEMDAS/BODMAS) and apply the rules of exponents correctly. We also need to be comfortable with the distributive property and combining like terms. Think of it as decluttering your math space – we want to organize and tidy up the expression.

Order of Operations (PEMDAS/BODMAS)

First off, let's talk about the order of operations, often remembered by the acronyms PEMDAS or BODMAS. This is our golden rule in simplifying expressions. It tells us the sequence in which we should perform operations:

• Parentheses / Brackets
• Exponents / Orders
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

So, if you see an expression with multiple operations, you know exactly where to start. It's like having a roadmap for your math journey. Following this order ensures that we all arrive at the same correct answer. Without it, math would be a free-for-all, and nobody wants that!

Rules of Exponents

Next up, let's brush up on the rules of exponents. Exponents are just a shorthand way of showing repeated multiplication. But when we start combining terms with exponents, things can get interesting. Here are a few key rules to keep in mind:

• Product of Powers: When multiplying like bases, add the exponents: aᵐ * aⁿ = aᵐ⁺ⁿ
• Power of a Power: When raising a power to another power, multiply the exponents: (aᵐ)ⁿ = aᵐⁿ
• Power of a Product: When raising a product to a power, distribute the exponent to each factor: (ab)ⁿ = aⁿbⁿ

These rules might seem like abstract formulas, but they're super useful in simplifying expressions. Mastering them is like unlocking a secret level in the algebra game. They help us manipulate expressions with confidence and make complex problems much more manageable.

Distributive Property

The distributive property is another crucial tool in our algebraic arsenal. It allows us to multiply a single term by a group of terms inside parentheses. The basic idea is that a(b + c) = ab + ac. You're essentially distributing the 'a' across the 'b' and the 'c'.

This property is particularly handy when dealing with expressions like 3(a²+b²)³ in our problem. We'll need to apply it (along with the rules of exponents) to expand and simplify this term. Think of the distributive property as a way to break down complex multiplications into simpler ones. It's like sharing the love (or the multiplication) equally!

Combining Like Terms

Lastly, combining like terms is all about tidying up our expressions. Like terms are terms that have the same variable raised to the same power. For example, 3x² and 5x² are like terms, but 3x² and 5x³ are not. We can combine like terms by adding or subtracting their coefficients (the numbers in front of the variables). So, 3x² + 5x² = 8x².

Combining like terms is like sorting your socks – you group the ones that are the same. It simplifies the expression and makes it easier to work with. After expanding and simplifying, we'll often need to combine like terms to get our final answer in its simplest form.

Step-by-Step Solution

Alright, with the basics covered, let's tackle the problem: (2a³) × 3(a²+b²)³ × 5b⁴. We'll break it down step by step so you can see exactly how it's done.

Step 1: Rearrange and Group Constants

The first thing we can do is rearrange the expression to group the constants together. This makes the multiplication of the numerical coefficients much clearer. We can rewrite the expression as:

(2 × 3 × 5) × a³ × (a²+b²)³ × b⁴

Multiplying the constants, 2, 3, and 5, gives us 30. So, the expression now looks like:

30 × a³ × (a²+b²)³ × b⁴

This simple rearrangement already makes the expression look a bit less daunting. It's like organizing your ingredients before you start cooking – everything is in its place, and you're ready to go!

Step 2: Expand (a²+b²)³

Next, we need to tackle the term (a²+b²)³. This means we're raising the binomial (a²+b²) to the power of 3. This is where we'll use the binomial theorem or simply multiply it out step by step. For simplicity and clarity, let's multiply it out. Remember, (a²+b²)³ = (a²+b²) × (a²+b²) × (a²+b²). We'll do this in stages.

First, let's multiply the first two factors:

(a²+b²) × (a²+b²) = a⁴ + 2a²b² + b⁴

Now, we multiply this result by the remaining (a²+b²):

(a⁴ + 2a²b² + b⁴) × (a²+b²) = a⁶ + 2a⁴b² + a²b⁴ + a⁴b² + 2a²b⁴ + b⁶

Combining like terms, we get:

a⁶ + 3a⁴b² + 3a²b⁴ + b⁶

This expansion might look a bit intimidating, but we've broken it down into manageable steps. Expanding this term is crucial because it allows us to see all the individual terms and combine them later if possible.

Step 3: Substitute the Expanded Form

Now that we've expanded (a²+b²)³, we can substitute it back into our expression. Remember, our expression was:

30 × a³ × (a²+b²)³ × b⁴

Substituting the expanded form, we get:

30 × a³ × (a⁶ + 3a⁴b² + 3a²b⁴ + b⁶) × b⁴

This is starting to look interesting! We've replaced a complex term with its expanded form, and now we can move on to the next step.

Step 4: Distribute a³ and b⁴

Now, we need to distribute a³ and b⁴ across the terms inside the parentheses. This means we'll multiply a³ by each term in (a⁶ + 3a⁴b² + 3a²b⁴ + b⁶) and then multiply the result by b⁴.

First, let's distribute a³:

a³ × (a⁶ + 3a⁴b² + 3a²b⁴ + b⁶) = a⁹ + 3a⁷b² + 3a⁵b⁴ + a³b⁶

Now, we multiply this by b⁴:

(a⁹ + 3a⁷b² + 3a⁵b⁴ + a³b⁶) × b⁴ = a⁹b⁴ + 3a⁷b⁶ + 3a⁵b⁸ + a³b¹⁰

So, our expression now looks like:

30 × (a⁹b⁴ + 3a⁷b⁶ + 3a⁵b⁸ + a³b¹⁰)

Distributing these terms is like delivering the multiplication to each member of the group. It ensures that each term is correctly accounted for.

Step 5: Distribute the Constant

Finally, we need to distribute the constant 30 across the terms inside the parentheses:

30 × (a⁹b⁴ + 3a⁷b⁶ + 3a⁵b⁸ + a³b¹⁰) = 30a⁹b⁴ + 90a⁷b⁶ + 90a⁵b⁸ + 30a³b¹⁰

This is the final step in our simplification journey. We've multiplied each term by 30, and we're left with a simplified expression.

The Final Simplified Expression

So, the simplified form of (2a³) × 3(a²+b²)³ × 5b⁴ is:

30a⁹b⁴ + 90a⁷b⁶ + 90a⁵b⁸ + 30a³b¹⁰

Common Mistakes to Avoid

Simplifying algebraic expressions can be tricky, and it's easy to make mistakes along the way. Let's look at some common pitfalls and how to avoid them.

Incorrect Order of Operations

One of the most common mistakes is not following the order of operations (PEMDAS/BODMAS). Always remember to do parentheses/brackets first, then exponents/orders, then multiplication and division (from left to right), and finally addition and subtraction (from left to right). Skipping steps or doing them in the wrong order can lead to incorrect answers.

Errors with Exponents

Exponent rules can be confusing if you don't apply them correctly. Remember, when multiplying like bases, you add the exponents, not multiply them. Similarly, when raising a power to a power, you multiply the exponents, not add them. Double-check your exponent rules to avoid these common errors.

Incorrect Distribution

The distributive property is a powerful tool, but it's easy to make mistakes if you're not careful. Make sure you multiply the term outside the parentheses by every term inside the parentheses. Don't leave anyone out! A common mistake is to only multiply by the first term and forget the others.

Not Combining Like Terms

After expanding and simplifying, always remember to combine like terms. This means adding or subtracting terms that have the same variable raised to the same power. Forgetting to do this can leave your expression unsimplified, which isn't the goal!

Sign Errors

Pay close attention to signs (positive and negative) throughout the simplification process. A simple sign error can throw off your entire answer. Keep track of your signs, especially when distributing negative terms.

Practice Problems

Now that we've walked through the solution and discussed common mistakes, it's time to put your skills to the test! Here are a few practice problems for you to try:

1. Simplify: (4x²) × 2(x+y)² × 3y³
2. Simplify: (a⁴) × 5(a²+b²)² × 2b²
3. Simplify: (3p³) × 4(p²+q²)³ × q⁵

Work through these problems step by step, just like we did in the example. Pay attention to the order of operations, exponent rules, distributive property, and combining like terms. The more you practice, the more confident you'll become in simplifying algebraic expressions.

Conclusion

Simplifying algebraic expressions might seem daunting at first, but with a systematic approach, it becomes much more manageable. We've shown you how to break down a complex problem into smaller steps, apply the rules of algebra, and avoid common mistakes. Remember, the key is to understand the basics, practice consistently, and take your time. You've got this!

So, guys, keep practicing, and soon you'll be simplifying algebraic expressions like a pro. Math can be challenging, but it's also incredibly rewarding. Keep pushing, keep learning, and keep simplifying! You're doing great!