Equivalent Expression Of (2)(a-2b)(a-2b)(18)

Hey guys! Let's dive into a fascinating mathematical exploration today. We're going to break down the expression (2)(a-2b)(a-2b)(18) and find its equivalent form. This might seem daunting at first, but don't worry, we'll take it step by step. Our main goal here is to make sure you not only understand the mechanics but also grasp the underlying concepts. So, grab your thinking caps, and let's get started!

Understanding the Basics of Algebraic Expressions

Before we jump into the specifics of our expression, it's crucial to have a solid understanding of algebraic expressions. An algebraic expression is essentially a combination of variables, constants, and mathematical operations like addition, subtraction, multiplication, and division. Variables are symbols (usually letters like 'a' and 'b') that represent unknown values, while constants are fixed numerical values (like 2 and 18 in our case). The beauty of algebraic expressions lies in their ability to represent a wide range of mathematical relationships in a concise and general way.

In our expression, (2)(a-2b)(a-2b)(18), we can identify several key components. We have the constants 2 and 18, the variable terms 'a' and 'b', and the expression (a-2b) which appears twice. This repetition is a significant clue that we might be dealing with a squared term. To simplify this expression, we'll be using the fundamental properties of arithmetic and algebra, such as the commutative, associative, and distributive properties. These properties allow us to rearrange and combine terms in a way that makes the expression easier to handle. For instance, the commutative property tells us that the order of multiplication doesn't matter (2 * 3 is the same as 3 * 2), while the distributive property helps us expand expressions like a(b + c) into ab + ac. By mastering these basics, we lay a strong foundation for tackling more complex algebraic manipulations. Remember, practice makes perfect, so the more you work with these concepts, the more comfortable you'll become.

Breaking Down the Expression (2)(a-2b)(a-2b)(18)

Let's start by focusing on the individual components of our expression, (2)(a-2b)(a-2b)(18). The first thing that might catch your eye is the repetition of the (a-2b) term. This immediately suggests that we can rewrite this part of the expression using an exponent. Specifically, (a-2b)(a-2b) is the same as (a-2b)². This is a crucial simplification step because it allows us to consolidate two identical terms into a single, more manageable term. Now our expression looks like this: 2 * (a-2b)² * 18.

Next, we can simplify the constants. We have the numbers 2 and 18, which are being multiplied together. Basic arithmetic tells us that 2 multiplied by 18 is 36. So, we can replace 2 * 18 with 36, and our expression becomes: 36 * (a-2b)². This is a significant simplification because we've reduced the number of individual terms and made the expression more compact. At this stage, we've handled the constants and the repeated term, leaving us with a single constant multiplied by a squared expression. This form is much easier to work with and sets us up for the next step, which involves expanding the squared term. Remember, simplifying expressions is all about breaking them down into smaller, more manageable parts and then using mathematical rules to combine those parts in a meaningful way.

Simplifying the Expression Step-by-Step

Now, let's walk through the simplification process step-by-step. Our starting point is the expression (2)(a-2b)(a-2b)(18). As we discussed earlier, the first key observation is the repetition of the (a-2b) term. This allows us to rewrite the expression as 2 * (a-2b)² * 18. This is a crucial step because it consolidates two terms into one, making the expression more manageable.

The next step involves simplifying the constants. We have 2 and 18 being multiplied together. Multiplying these constants gives us 2 * 18 = 36. So, we can replace 2 * 18 with 36, and our expression now looks like this: 36 * (a-2b)². We've now simplified the constants, leaving us with a single constant multiplied by a squared expression. This form is much cleaner and easier to work with.

The final step is to expand the squared term, (a-2b)². Remember that squaring an expression means multiplying it by itself. So, (a-2b)² is the same as (a-2b)(a-2b). To expand this, we'll use the distributive property (also known as the FOIL method for binomials). This involves multiplying each term in the first parenthesis by each term in the second parenthesis. Specifically, we have: (a-2b)(a-2b) = aa + a(-2b) + (-2b)a + (-2b)(-2b). Simplifying this gives us a² - 2ab - 2ab + 4b². Combining like terms (-2ab and -2ab) results in a² - 4ab + 4b². So, (a-2b)² expands to a² - 4ab + 4b².

Now we substitute this expanded form back into our expression: 36 * (a² - 4ab + 4b²). The final step is to distribute the 36 across the terms inside the parenthesis. This means multiplying each term inside the parenthesis by 36. So, we have: 36 * a² - 36 * 4ab + 36 * 4b². Performing these multiplications gives us 36a² - 144ab + 144b². And there you have it! The fully simplified and equivalent expression is 36a² - 144ab + 144b².

Expanding the Squared Term (a-2b)²

Let's take a closer look at expanding the squared term (a-2b)². This is a crucial step in simplifying our original expression, and it's a technique that comes up frequently in algebra. Squaring an expression, as we've mentioned, means multiplying it by itself. So, (a-2b)² is the same as (a-2b)(a-2b). To expand this product, we use the distributive property, which is often remembered by the acronym FOIL (First, Outer, Inner, Last). This method ensures that we multiply each term in the first parenthesis by each term in the second parenthesis.

Following the FOIL method, we first multiply the First terms: a * a = a². Then, we multiply the Outer terms: a * (-2b) = -2ab. Next, we multiply the Inner terms: (-2b) * a = -2ab. Finally, we multiply the Last terms: (-2b) * (-2b) = 4b². So, the expansion looks like this: a² - 2ab - 2ab + 4b². The next step is to combine like terms. In this case, we have two terms that are '-2ab'. Combining these gives us -2ab - 2ab = -4ab. This simplifies our expression to a² - 4ab + 4b². This is the expanded form of (a-2b)², and it's a trinomial (an expression with three terms). Understanding how to expand squared binomials like this is a fundamental skill in algebra, and it's essential for simplifying more complex expressions and solving equations. Remember, the FOIL method is a handy tool, but it's really just an application of the distributive property. The more you practice, the more comfortable you'll become with this technique, and you'll be able to expand these expressions with ease.

Final Simplified Expression: 36a² - 144ab + 144b²

After carefully breaking down and simplifying the original expression (2)(a-2b)(a-2b)(18), we've arrived at the final simplified form: 36a² - 144ab + 144b². This expression is equivalent to the original but is much easier to understand and work with. Let's recap the steps we took to get here. First, we recognized the repeated (a-2b) term and rewrote it as (a-2b)². Then, we simplified the constants 2 and 18 by multiplying them together to get 36. This gave us 36 * (a-2b)². Next, we expanded the squared term (a-2b)² using the distributive property (or the FOIL method), which resulted in a² - 4ab + 4b². Finally, we distributed the 36 across the terms in the trinomial, multiplying each term by 36 to get our final simplified expression: 36a² - 144ab + 144b².

This final expression is a quadratic trinomial, meaning it's a polynomial with three terms and the highest power of the variable is 2. The terms are 36a² (a squared term), -144ab (a product of variables term), and 144b² (another squared term). This form is often the most convenient for further algebraic manipulations, such as solving equations or graphing functions. By simplifying the original expression, we've not only made it more compact but also revealed its underlying structure. This is a key goal in algebra – to transform expressions into forms that make their properties and relationships more apparent. Remember, simplification isn't just about getting a shorter answer; it's about gaining a deeper understanding of the mathematical expression and its behavior. So, next time you encounter a complex expression, remember these steps, and you'll be well-equipped to tackle it!

Conclusion: Mastering Algebraic Simplification

Alright guys, we've reached the end of our journey through simplifying the expression (2)(a-2b)(a-2b)(18). We started with a seemingly complex expression and, through careful step-by-step simplification, arrived at the equivalent form: 36a² - 144ab + 144b². This process highlights the power and elegance of algebraic manipulation. By applying fundamental properties and techniques, we can transform expressions into more manageable and insightful forms.

The key takeaways from this exploration are the importance of recognizing repeated terms, simplifying constants, expanding squared terms, and using the distributive property. These are essential tools in your algebraic toolkit, and mastering them will significantly enhance your ability to tackle a wide range of mathematical problems. Remember, algebraic simplification isn't just about finding the right answer; it's about developing a deeper understanding of mathematical relationships and building problem-solving skills.

So, keep practicing, keep exploring, and don't be afraid to tackle complex expressions. With a solid foundation in these basic principles, you'll be well on your way to becoming an algebraic whiz! And hey, if you ever get stuck, just remember the steps we've covered here, and you'll be able to break down even the trickiest expressions. Keep up the great work, and happy simplifying!