Simplifying Algebraic Expressions: A Step-by-Step Guide

by ADMIN 56 views

Hey guys! Today, let's dive into simplifying a fun algebraic expression. We'll break down the problem step-by-step, so even if algebra makes you sweat a little, you'll be a pro by the end of this. Our mission, should we choose to accept it, is to simplify this expression:

(4k^2 - 100) / (5k^2 - 80) * (4k + 16) / (4k - 20)

Sounds intimidating, right? Don't worry, we've got this! Let’s jump in!

1. Factoring: The Key to Simplification

In simplifying algebraic expressions, the first crucial step involves factoring. Think of factoring as the secret weapon in your algebra arsenal. It allows us to break down complex expressions into simpler, manageable components. By identifying common factors, we can rewrite the terms, paving the way for cancellations and simplifications that would otherwise be hidden. Factoring isn't just a mechanical process; it's about recognizing patterns and applying algebraic identities to transform expressions into a form where their underlying structure becomes clear. This initial step sets the stage for the rest of the simplification process, making it much easier to identify and eliminate redundancies. So, let's roll up our sleeves and get factoring; it’s the magic that makes algebra less like a puzzle and more like a smooth ride.

Factoring 4k^2 - 100

Okay, let's tackle the first part: 4k^2 - 100. Notice anything special? This looks like a difference of squares! Remember the formula: a^2 - b^2 = (a + b)(a - b). Our expression fits this pattern perfectly.

  • We can rewrite 4k^2 as (2k)^2 and 100 as 10^2.
  • So, 4k^2 - 100 becomes (2k + 10)(2k - 10).
  • But wait, there's more! We can factor out a 2 from each term: 2(k + 5) * 2(k - 5).
  • This simplifies to 4(k + 5)(k - 5). Awesome!

Factoring 5k^2 - 80

Next up, let's factor 5k^2 - 80. First, we can factor out a common factor of 5:

  • 5k^2 - 80 = 5(k^2 - 16).
  • Again, we spot a difference of squares! k^2 - 16 is the same as k^2 - 4^2.
  • Using our formula, this factors to (k + 4)(k - 4).
  • Putting it all together, 5k^2 - 80 becomes 5(k + 4)(k - 4). You're getting the hang of it!

Factoring 4k + 16

This one's a bit simpler. We just need to find the greatest common factor (GCF) of 4k and 16:

  • The GCF is 4, so we factor it out: 4(k + 4). Easy peasy!

Factoring 4k - 20

Almost there! Let's factor 4k - 20:

  • Again, the GCF is 4, so we factor it out: 4(k - 5). You're a factoring machine!

2. Rewriting the Expression: Putting it All Together

Now that we've masterfully factored each part, the next step is to rewrite the entire expression using our factored forms. This is where things start to get really satisfying because we're setting the stage for some serious simplification. By substituting the original terms with their factored equivalents, we transform the expression into a form that's much easier to manipulate. This step is like laying out all the pieces of a puzzle; you can finally see how they connect and what needs to be done to complete the picture. So, let's take those factored expressions and slot them back into the original equation. Get ready to see how this transformation makes the complexity melt away, leaving us with a clear path to the final, simplified answer. It’s like magic, but it’s actually just clever algebra!

Let's replace the original terms with their factored forms:

  • (4k^2 - 100) / (5k^2 - 80) * (4k + 16) / (4k - 20) becomes
  • [4(k + 5)(k - 5)] / [5(k + 4)(k - 4)] * [4(k + 4)] / [4(k - 5)]

See how much cleaner it looks already? All those factors are just begging to be simplified!

3. Cancellation: The Art of Strategic Elimination

Here's where the fun really begins! Cancellation in algebra is like the satisfying click of puzzle pieces falling into place. It's the process of identifying and eliminating common factors from the numerator and the denominator of our expression, and it’s a crucial step in simplification. Think of it as trimming away the excess to reveal the core, simplified form. This step isn't just about crossing things out; it's about recognizing the structure of the expression and using the rules of division to reduce it to its simplest terms. Each cancellation brings us closer to the final answer, making the entire process feel incredibly rewarding. So, sharpen your algebraic vision and let's get to work, strategically eliminating those common factors and watching the expression shrink into something beautifully simple.

Now, let's cancel out common factors. Remember, anything that appears in both the numerator (top) and the denominator (bottom) can be canceled:

  • We have a 4 in the numerator and denominator – poof, gone!
  • We also have (k + 4) in both – bye-bye!
  • And look, (k - 5) appears in both too – adios!

Our expression now looks like this:

  • [4(k + 5)(k - 5)] / [5(k + 4)(k - 4)] * [4(k + 4)] / [4(k - 5)] simplifies to
  • [4(k + 5)] / [5(k - 4)]

We've eliminated so much already! It's like magic, but it's just algebra. You're doing great!

4. Final Simplification: The Home Stretch

We're almost there! The final simplification step is where we tie up any loose ends and present our answer in its most elegant form. After all the factoring and cancellation, this is the moment where we make sure everything is as clean and concise as possible. It might involve distributing any remaining terms or combining like terms to ensure our expression is truly in its simplest state. This isn't just about getting the right answer; it's about presenting it with clarity and precision. Think of it as the final polish on a masterpiece, ensuring it shines in all its simplified glory. So, let's take one last look at our expression, make those final adjustments, and proudly present our simplified answer to the world. You've come this far; let's make it perfect!

Let's take a look at what we have left:

  • [4(k + 5)] / [5(k - 4)]

There are no more common factors to cancel, and we can leave it just like this, or distribute the constants if we prefer. Both answers are perfectly correct!

Option 1: Leaving it factored

  • [4(k + 5)] / [5(k - 4)]

Option 2: Distributing

  • (4k + 20) / (5k - 20)

5. The Final Answer

So, the fully simplified form of the expression

(4k^2 - 100) / (5k^2 - 80) * (4k + 16) / (4k - 20)

is either:

[4(k + 5)] / [5(k - 4)]

Or:

(4k + 20) / (5k - 20)

You did it! Give yourself a pat on the back. You've successfully simplified a complex algebraic expression. Remember, the key is to break it down into smaller steps: factoring, rewriting, canceling, and simplifying. Keep practicing, and you'll become an algebra whiz in no time!