Unveiling The Product: Expanding $(3a^4 + 4)^2$

Hey math enthusiasts! Today, we're diving into a classic algebra problem: finding the product of the expression $\left(3 a^4+4\right)^2$. Don't worry, it's not as scary as it looks! We're gonna break it down step-by-step, making sure everyone understands the process. This is a fundamental concept, and once you grasp it, you'll be able to tackle similar problems with ease. So, buckle up, grab your pencils, and let's get started on this exciting mathematical journey! We'll explore the power of the binomial expansion and how to simplify these algebraic expressions. This skill is critical for further studies in mathematics, so let's make sure we build a strong foundation! The main aim here is to simplify and expand the expression using different methods, and finally, present the answer. By the end, you'll feel confident in expanding binomials and simplifying algebraic expressions. This type of problem often shows up in various exams and quizzes, and therefore, it is important to practice. Let's make sure that our mathematical skills are sharp. Let's start with a basic understanding of what the question is asking us to do. This question basically is asking us to find the product of $(3a^4+4)$ multiplied by itself. Let's simplify and make the calculations easy. Are you guys ready? Let's begin the exciting journey!

Decoding the Expression: $(3a^4 + 4)^2$

Alright, let's break down this expression piece by piece. First off, what does the exponent '2' mean when it's sitting outside of the parentheses? It essentially means we're multiplying the entire expression inside the parentheses by itself. So, $\left(3 a^4+4\right)^2$ is the same as $\left(3 a^4+4\right) \times \left(3 a^4+4\right)$. Got it? Awesome! Now, let's look at what's inside the parentheses: we have a binomial, which is just a fancy way of saying an expression with two terms. In this case, our terms are $3a^4$ and $4$. The goal is to multiply this binomial by itself. This is where the fun starts! We can solve this problem using a couple of methods. First, we'll go through the direct multiplication method, which involves distributing each term of the first binomial with each term of the second. The second one will use the formula $(a + b)^2 = a^2 + 2ab + b^2$ for the sake of simplicity. Both will lead us to the same solution. So, let's start with the direct multiplication method. Direct multiplication is a straightforward way to expand the expression. It involves distributing each term of the first binomial to each term of the second binomial and then simplifying. For example, $(A + B)(C + D) = A*C + A*D + B*C + B*D$. We can apply this method and solve the question now. Therefore, this method is useful and easy to understand. Ready to jump in?

Method 1: Direct Multiplication

Let's apply the distributive property. We will multiply each term of the first binomial by each term of the second binomial.

$\left(3 a^4+4\right) \times \left(3 a^4+4\right)$

• Multiply $3a^4$ by $3a^4$: $(3a^4) * (3a^4) = 9a^8$ (Remember, when multiplying exponents, we add them: $4 + 4 = 8$).
• Multiply $3a^4$ by $4$: $(3a^4) * 4 = 12a^4$
• Multiply $4$ by $3a^4$: $4 * (3a^4) = 12a^4$
• Multiply $4$ by $4$: $4 * 4 = 16$

Now, let's put it all together. Adding all these results, we get:

$9a^8 + 12a^4 + 12a^4 + 16$

Finally, combine the like terms (the terms with the same variable and exponent). In this case, we have two $12a^4$ terms:

$9a^8 + (12a^4 + 12a^4) + 16$

Which simplifies to:

$9a^8 + 24a^4 + 16$

And that's our answer, guys! We've found the product using direct multiplication! Not too bad, huh?

Method 2: Using the $(a + b)^2$ Formula

Alright, let's try another approach. There's a handy formula we can use when we have a binomial squared: $(a + b)^2 = a^2 + 2ab + b^2$. This formula is a shortcut that can save us a bit of time. If you remember the formula, it can make problems like these a breeze! Let's identify the 'a' and 'b' in our expression, $\left(3 a^4+4\right)^2$. In this case, $a = 3a^4$ and $b = 4$. Now, let's plug these values into our formula. The steps are easy to remember. We square the first term, add twice the product of both terms, and add the square of the second term.

So, according to the formula: $(3a^4 + 4)^2 = (3a^4)^2 + 2 * (3a^4) * 4 + 4^2$. Let's break it down further:

• $(3a^4)^2 = 9a^8$ (Square both the coefficient and the variable term)
• $2 * (3a^4) * 4 = 24a^4$
• $4^2 = 16$

Putting it all together, we get: $9a^8 + 24a^4 + 16$. See, we get the same answer as before! The benefit of using this formula is that it simplifies the expansion process and allows you to find the answer faster. It saves you from having to do multiple steps, such as distributing terms one by one. This is also important because it can help with your problem-solving abilities. It helps you recognize patterns in problems.

Conclusion: The Final Product

Finding the product of $\left(3 a^4+4\right)^2$ is actually pretty simple. Whether you use direct multiplication or the formula $(a + b)^2$, you'll arrive at the same answer: $9a^8 + 24a^4 + 16$. Both of these methods are completely valid, so feel free to use whichever one you're more comfortable with. Practicing these kinds of problems will help you get better and more confident in your math skills! Remember, the key is to understand the concepts and practice consistently. Keep practicing, and you'll become a pro at expanding and simplifying algebraic expressions. You've got this! Now, go forth and conquer those algebra problems! This process involves a combination of understanding exponents, applying the distributive property, and combining like terms. Being able to do this will help you improve your mathematical capabilities.

Summary

In this article, we've explored the expansion of the binomial $\left(3 a^4+4\right)^2$. We utilized two primary methods to arrive at the solution. Firstly, direct multiplication, which involves a step-by-step distribution of terms. Secondly, we employed the formula $(a+b)^2 = a^2 + 2ab + b^2$ for a more streamlined approach. Both methods have proven to be effective and have yielded the same final answer.

The final product of $\left(3 a^4+4\right)^2$ is $9a^8 + 24a^4 + 16$. The ability to expand and simplify such expressions forms the cornerstone of algebraic manipulation and is crucial for advanced mathematical concepts. Consistent practice and a solid grasp of these fundamental principles will significantly enhance your problem-solving skills in mathematics.