Rectangle Perimeter: Spotting The Calculation Mistake

Hey guys! Ever stumbled upon a math problem that looks straightforward but has a sneaky little twist? Let's dive into one today, focusing on rectangle perimeters and radical simplification. We'll break down a student's attempt to calculate the perimeter of a rectangle and pinpoint exactly where the misstep occurred. This isn't just about finding the right answer; it's about understanding the process and avoiding common pitfalls. So, grab your thinking caps, and let's get started!

The Problem: Unveiling the Rectangle's Dimensions

Our mission, should we choose to accept it, involves a rectangle with some intriguing dimensions. The length is given as √50a³b², and the width is √200a³. Now, right off the bat, these expressions look a bit…intimidating. But don't worry! We're going to tackle them step by step. The core concept here is finding the perimeter, which, as you might recall, is simply the total distance around the rectangle. Remember that the formula for the perimeter of a rectangle is P = 2l + 2w, where 'l' represents the length and 'w' represents the width. Our challenge lies in simplifying those radical expressions before we can plug them into the formula. This involves understanding how to extract perfect squares from under the radical sign. It’s like a mathematical treasure hunt where we’re searching for pairs of factors that can be brought outside the square root. Think of it as decluttering the radicals to make the calculations smoother and more manageable. Before we dissect the student's solution, let's refresh our understanding of simplifying radicals. This is a crucial skill for tackling problems like this, and a solid grasp of the basics will make the rest of the process much clearer. We'll need this skill to accurately calculate the perimeter, so let's make sure we're all on the same page!

The Student's Attempt: A Step-by-Step Breakdown

A student, enthusiastic to solve this, approached the problem with the following steps:

2√50a³b² + 2√200a³ = 2 ⋅ 5ab√2a + 2 ⋅ 10a√2a
= 10ab√2a + 20a

At first glance, it might seem like they're on the right track, but there's a subtle yet significant error lurking within these calculations. Our job is to play detective and figure out exactly where the mistake crept in. We'll meticulously examine each step, comparing it to the correct simplification process. This isn't about pointing fingers; it's about learning from mistakes. By understanding where the student went astray, we can reinforce our own understanding and avoid similar errors in the future. We'll focus on the properties of square roots and how they interact with multiplication and addition. This is where the devil often hides in mathematical details. Keep a keen eye on how the student handled the coefficients and variables both inside and outside the radical signs. Did they correctly identify and extract perfect squares? Did they maintain the integrity of the expressions throughout the simplification process? These are the questions we need to answer to pinpoint the exact location of the error.

Spotting the Mistake: Where Did It Go Wrong?

Alright, let's put on our detective hats and carefully examine the student's solution. The initial step seems promising, with the student attempting to simplify the radicals. However, the devil is in the details! Let's break it down:

• Step 1: 2√50a³b² + 2√200a³ = 2 ⋅ 5ab√2a + 2 ⋅ 10a√2a

  Here's where the first red flag appears. While the simplification of 2√50a³b² to 2 ⋅ 5ab√2a looks correct (50 = 25 * 2, a³ = a² * a, and b² is a perfect square), the simplification of 2√200a³ to 2 ⋅ 10a√2a needs a closer look. While it's true that 200 = 100 * 2 and a³ = a² * a, the student seems to have stopped prematurely. Remember, we want to extract all perfect square factors. We need to dig a little deeper into that √200a³ term. Think about the largest perfect square that divides 200. Is 100 the only one? Or is there a more complete factorization we can achieve? This is the crucial point where the error originates. It's not a massive blunder, but it's enough to throw off the final answer. Let's make sure we nail this down before moving on.

• Step 2: = 10ab√2a + 20a

  This step carries forward the error from the previous step. The student combines the simplified terms, but because the initial simplification of 2√200a³ was incomplete, the final expression is incorrect. This highlights the importance of accuracy in each step of a mathematical process. A small mistake early on can snowball into a significant error later. It’s like a chain reaction – one wrong link breaks the whole chain. This also emphasizes the value of double-checking our work and ensuring that each simplification is as thorough as possible. Now, let’s roll up our sleeves and correct this mistake, showing the proper way to simplify and combine the terms.

The Correct Solution: A Step-by-Step Guide

Okay, guys, let's ditch the errors and find the true perimeter! We'll go through the correct simplification process step-by-step, making sure every detail is crystal clear. This is where we transform those intimidating radical expressions into something much more manageable. Remember, the key is to identify and extract those perfect square factors. Think of it as unlocking the hidden potential within the radicals. And don't worry, we'll take it slow and steady, leaving no room for confusion. By the end of this, you'll be a radical simplification pro!

1. Simplify √50a³b²:

   • Break down 50 into 25 * 2. Remember, 25 is a perfect square (5²).
   • Break down a³ into a² * a. a² is a perfect square.
   • b² is already a perfect square.
   • So, √50a³b² = √(25 * 2 * a² * a * b²) = 5ab√2a. This part the student actually got right!

2. Simplify √200a³:

   • Break down 200 into 100 * 2. 100 is a perfect square (10²).
   • Break down a³ into a² * a. a² is a perfect square.
   • So, √200a³ = √(100 * 2 * a² * a) = 10a√2a. Again, the student got this part right in their initial simplification... almost! The key is what they did with this term next.

3. Calculate the Perimeter (P = 2l + 2w):

   • P = 2(5ab√2a) + 2(10a√2a)
   • P = 10ab√2a + 20a√2a Here's where the student's solution started to go wrong! They dropped the √2a from the second term.

4. Combine Like Terms:

   • Notice that both terms now have a common factor of √2a. This is crucial! We can combine them just like we combine 'x' terms in algebra.
   • P = (10ab + 20a)√2a This is the correct final form.

See the difference? The correct answer includes the √2a in the second term, which significantly changes the result. This highlights the importance of paying attention to every single component of an expression during simplification. Missing a small detail like a radical can lead to a completely different outcome.

Key Takeaways: Avoiding Future Mistakes

So, what have we learned from this mathematical adventure? Let's recap the key takeaways to solidify our understanding and prevent similar errors in the future.

• Simplify Radicals Completely: Always ensure you've extracted all perfect square factors from under the radical. Don't stop at the first perfect square you find; keep digging until you can't simplify any further. Think of it like cleaning your room – you don't just tidy up the obvious clutter; you look under the bed and in the closet too! This thorough approach is essential for accurate simplification.
• Pay Attention to Detail: Math is a precise language. Every symbol, every number, every radical matters. Make sure you're meticulously copying and manipulating each term. It’s like following a recipe – miss one ingredient or misread the instructions, and the final dish won't turn out right. Double-checking your work can be a lifesaver here.
• Combine Like Terms Correctly: You can only combine terms that have the exact same radical part. This is similar to combining 'x' terms in algebra. If the radicals are different, you can't simply add the coefficients. It’s like trying to add apples and oranges – they're both fruit, but you can't say you have a combined quantity of “apple-oranges.”
• Practice Makes Perfect: The more you practice simplifying radicals and working with perimeters, the more comfortable and confident you'll become. It's like learning a musical instrument – the more you practice, the better you get. So, don't be afraid to tackle more problems and challenge yourself!

By understanding these key principles, you'll be well-equipped to handle similar problems with ease. Remember, math isn't just about memorizing formulas; it's about developing a solid understanding of the underlying concepts. And with practice and attention to detail, you can conquer any mathematical challenge!