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

by Aria Freeman 43 views

Hey there, math enthusiasts! Let's dive into the fascinating world of algebraic expressions and tackle a problem that might seem a bit daunting at first glance. We're going to break down the expression (2)(a-2b)(a-2b)(18), explore its components, and ultimately find an equivalent expression that's easier to work with. So, buckle up and get ready for a journey into the realm of mathematical simplification!

Understanding the Expression

Before we jump into simplifying, let's make sure we truly grok what this expression is all about. At its heart, it's a product of several factors. We've got constants like 2 and 18, and we've also got the binomial expression (a-2b) multiplied by itself. This little detail is a crucial clue, guys, because it hints at the possibility of a squared term lurking beneath the surface.

Let's break it down piece by piece:

  • The (a-2b) term is a binomial, meaning it's an expression with two terms (a and -2b). These terms are connected by a subtraction operation.
  • The fact that (a-2b) appears twice means we're essentially squaring this binomial, which is something we'll definitely want to keep in mind as we simplify.
  • The constants 2 and 18 are just regular numbers, and we know from the basic rules of arithmetic that we can multiply them together.

So, with this foundational understanding in place, we're well-equipped to start our simplification adventure. Think of it like untangling a knot – we just need to find the right approach to loosen the strands and reveal the simpler form.

The Power of Simplification

Now, you might be wondering, why even bother simplifying? Well, in mathematics, simplicity often equates to clarity and ease of use. A simplified expression is usually:

  • Easier to evaluate: When you need to plug in values for the variables, a simpler expression means fewer steps and a lower chance of making errors.
  • Easier to manipulate: Simplified expressions are much more manageable when you need to solve equations, graph functions, or perform other mathematical operations.
  • More revealing: Sometimes, the true nature of a mathematical relationship is obscured by a complex expression. Simplification can help us see the underlying structure and patterns more clearly.

In this particular case, simplifying (2)(a-2b)(a-2b)(18) will not only make it more compact but might also reveal some interesting algebraic forms. We're on the hunt for an equivalent expression – one that represents the same mathematical relationship but does so in a more streamlined way.

Step-by-Step Simplification

Alright, let's get our hands dirty and walk through the simplification process step by step. We'll use the fundamental principles of algebra to transform our original expression into its simpler equivalent.

Step 1: Combine the Constants

Our expression starts with (2)(a-2b)(a-2b)(18). The first thing we can tackle is those constants, 2 and 18. Multiplication is commutative, meaning we can multiply numbers in any order. So, let's bring those constants together:

(2)(18)(a-2b)(a-2b)

Now, we simply multiply 2 and 18:

36(a-2b)(a-2b)

This is already looking cleaner, right? We've reduced two constants to a single constant, 36.

Step 2: Recognize the Squared Binomial

Now comes the crucial observation: we have (a-2b) multiplied by itself. This is the very definition of squaring an expression. So, we can rewrite (a-2b)(a-2b) as (a-2b)^2. This gives us:

36(a-2b)^2

This is a significant step forward. We've transformed the repeated binomial multiplication into a concise squared term. But, we're not quite done yet. We can still expand that squared binomial.

Step 3: Expand the Squared Binomial

To expand (a-2b)^2, we need to remember the pattern for squaring a binomial. There are a couple of ways to think about this:

  • The FOIL Method: If you remember the acronym FOIL (First, Outer, Inner, Last), this is a handy way to multiply two binomials. We're essentially multiplying (a-2b) by (a-2b). So, we multiply:

    • First terms: a * a = a^2
    • Outer terms: a * -2b = -2ab
    • Inner terms: -2b * a = -2ab
    • Last terms: -2b * -2b = 4b^2

    Then, we add these results together: a^2 - 2ab - 2ab + 4b^2

  • The Binomial Square Formula: There's a handy formula that directly tells us how to square a binomial: (x - y)^2 = x^2 - 2xy + y^2. In our case, x is a and y is 2b. So, applying the formula:

    • (a - 2b)^2 = a^2 - 2(a)(2b) + (2b)^2 = a^2 - 4ab + 4b^2

Notice that both methods lead us to the same result: a^2 - 4ab + 4b^2. So, we can replace (a-2b)^2 with this expanded form:

36(a^2 - 4ab + 4b^2)

Step 4: Distribute the Constant

We're almost there! The last step is to distribute the 36 across the terms inside the parentheses. This means multiplying each term in the trinomial by 36:

  • 36 * a^2 = 36a^2
  • 36 * -4ab = -144ab
  • 36 * 4b^2 = 144b^2

Putting it all together, we get our final simplified expression:

36a^2 - 144ab + 144b^2

The Equivalent Expression Unveiled

So, after our step-by-step journey through simplification, we've arrived at the equivalent expression for (2)(a-2b)(a-2b)(18): 36a^2 - 144ab + 144b^2.

Isn't it amazing how a seemingly complex expression can be transformed into something so much cleaner and more manageable? This simplified form is not only easier to work with but also reveals the underlying structure of the mathematical relationship. We can now clearly see the quadratic nature of the expression, with its squared terms and the interaction between a and b.

Key Takeaways

Before we wrap up, let's highlight the key concepts and techniques we used in this simplification adventure:

  • Combining Constants: Multiplying constants together is a fundamental step in simplification.
  • Recognizing Squared Binomials: Spotting patterns like (x-y)^2 allows us to use powerful algebraic shortcuts.
  • Expanding Squared Binomials: We can use the FOIL method or the binomial square formula to expand these terms efficiently.
  • Distributive Property: Distributing a constant across a sum or difference is essential for fully simplifying expressions.

By mastering these techniques, you'll be well-equipped to tackle a wide range of algebraic simplification problems. Remember, practice makes perfect, so keep exploring and experimenting with different expressions!

Real-World Applications

Now, you might be thinking,