Simplifying The Expression (3a^4 - 5a^3) * 4a^5 A Step-by-Step Guide

Hey guys! Ever stumbled upon an algebraic expression that looks like it belongs in a math maze? Today, we're going to break down one of those expressions and make it super simple. We're tackling (3a^4 - 5a^3) * 4a^5. Don't worry, it's not as scary as it looks! We'll go through each step together, so by the end of this, you'll be simplifying expressions like a pro. Let's dive in and demystify this mathematical puzzle!

Understanding the Basics: The Distributive Property and Exponent Rules

Before we jump into the main problem, let's quickly refresh some fundamental concepts. Think of these as the tools in our mathematical toolbox. We'll be using the distributive property and the rules of exponents, so let's make sure we're on the same page.

The Distributive Property: Spreading the Love

The distributive property is like sharing the wealth in an expression. It states that a(b + c) = ab + ac. In simpler terms, if you have a term outside parentheses multiplied by a sum (or difference) inside the parentheses, you need to multiply the outside term by each term inside. Imagine you're hosting a party (the 'a' outside the parentheses), and you need to give a party favor to each guest (the 'b' and 'c' inside the parentheses). You wouldn't just give it to one guest, right? You'd distribute the love!

In our expression, the 4a^5 is playing the role of the host, and the (3a^4 - 5a^3) are our guests. We'll need to make sure 4a^5 gets multiplied by both 3a^4 and -5a^3.

Rules of Exponents: Powering Up

Next up, exponents! Remember that an exponent tells you how many times to multiply a number (or variable) by itself. For example, a^4 means a * a * a * a. When we multiply terms with the same base (like 'a' in our case), we add the exponents. This is the crucial rule we'll use: a^m * a^n = a^(m+n).

Think of it like this: If you have a^2 (a * a) and you multiply it by a^3 (a * a * a), you end up with a total of five 'a's multiplied together (a * a * a * a * a), which is a^5. So, when we see a^4 * a^5, we know we're going to add those exponents.

With these tools in our toolbox, we're ready to tackle the expression! Let's get to the step-by-step simplification.

Step-by-Step Simplification of (3a^4 - 5a^3) * 4a^5

Okay, let's break down this expression piece by piece. We're going to take it one step at a time, so it's super clear how we arrive at the final answer. Remember, the key is to stay organized and apply the rules we just discussed.

Step 1: Apply the Distributive Property

First, we'll use the distributive property to multiply the 4a^5 term by each term inside the parentheses. This means we'll multiply 4a^5 by 3a^4 and then multiply 4a^5 by -5a^3. It's like we're expanding the expression, making it easier to work with.

So, we get:

(4a^5 * 3a^4) + (4a^5 * -5a^3)

Notice how we've essentially split the original expression into two separate multiplication problems. This is a crucial step because it allows us to focus on each part individually.

Step 2: Multiply the Coefficients

Now, let's focus on those coefficients! Coefficients are just the numbers in front of the variables. In our expression, we have 4 * 3 in the first part and 4 * -5 in the second part. Multiplying these gives us:

12a^5 * a^4 + (-20)a^5 * a^3

We've taken care of the numerical part of the multiplication. Now, it's time to deal with the variables and their exponents.

Step 3: Apply the Exponent Rule for Multiplication

This is where the magic happens! Remember the rule: a^m * a^n = a^(m+n). We're going to use this to combine the 'a' terms in each part of our expression. In the first part, we have a^5 * a^4, and in the second part, we have a^5 * a^3. Let's add those exponents:

12a^(5+4) + (-20)a^(5+3)

This simplifies to:

12a^9 + (-20)a^8

We've successfully combined the 'a' terms! We're almost there.

Step 4: Simplify the Expression

Finally, let's clean up our expression. We have a positive term and a negative term, so we can rewrite the expression as:

12a^9 - 20a^8

And that's it! We've simplified the expression (3a^4 - 5a^3) * 4a^5 to 12a^9 - 20a^8. Give yourself a pat on the back! You've just navigated a potentially tricky algebraic expression.

Common Mistakes to Avoid

Simplifying expressions can be a bit like navigating a maze, and it's easy to make a wrong turn. Let's highlight some common pitfalls to watch out for so you can avoid them.

Forgetting to Distribute

This is a classic mistake! Remember, the distributive property means you need to multiply the term outside the parentheses by every term inside. It's like making sure everyone gets a slice of pizza. If you forget to distribute, you'll end up with an incorrect answer.

Example of the mistake:

Incorrect: 4a^5 * (3a^4 - 5a^3) = 12a^9 - 5a^3 (Forgot to multiply 4a^5 by -5a^3)

Correct: 4a^5 * (3a^4 - 5a^3) = 12a^9 - 20a^8

Incorrectly Applying Exponent Rules

The rules of exponents are powerful, but they need to be used correctly. The most common mistake is adding exponents when you shouldn't or forgetting to add them when you should.

Example of the mistake:

Incorrect: a^5 * a^4 = a^20 (Multiplied exponents instead of adding)

Correct: a^5 * a^4 = a^9

Sign Errors

Watch out for those pesky negative signs! They can easily trip you up. Pay close attention when multiplying negative numbers and make sure you carry the negative sign correctly throughout your calculations.

Example of the mistake:

Incorrect: 4a^5 * -5a^3 = 20a^8 (Forgot the negative sign)

Correct: 4a^5 * -5a^3 = -20a^8

Not Simplifying Completely

Sometimes, you might do most of the work correctly but forget to simplify the final expression. Make sure you've combined all like terms and that your answer is in its simplest form.

By keeping these common mistakes in mind, you'll be well-equipped to simplify expressions accurately and confidently!

Practice Problems: Test Your Skills!

Alright, now that we've walked through the solution and highlighted common mistakes, it's time to put your knowledge to the test! Practice is key to mastering any skill, and simplifying expressions is no exception. So, let's dive into a few practice problems.

Problem 1:

Simplify: (2x^3 + 7x^2) * 3x^4

Problem 2:

Simplify: (5b^2 - 3b^5) * 2b^3

Problem 3:

Simplify: (-4y^4 + 2y^2) * 5y^6

Work through these problems step-by-step, just like we did in the example. Remember to apply the distributive property, multiply the coefficients, and use the rules of exponents correctly. Don't forget to watch out for those negative signs!

If you get stuck, don't worry! Go back and review the steps we covered earlier in this guide. The more you practice, the more comfortable you'll become with simplifying expressions.

Conclusion: You've Got This!

So, guys, we've journeyed through the simplification of the expression (3a^4 - 5a^3) * 4a^5, and you've learned some valuable skills along the way! We started by understanding the distributive property and exponent rules, then we broke down the problem step-by-step. We even highlighted common mistakes to avoid and gave you some practice problems to sharpen your skills.

Remember, simplifying expressions is like building with LEGOs. Each step is a block, and when you put them together correctly, you create something awesome! Don't be afraid to take it slow, review the basics, and practice consistently. With a little effort, you'll be simplifying algebraic expressions like a mathematical maestro!

Keep practicing, stay curious, and most importantly, have fun with math! You've got this!