Simplifying $7 \sqrt{8} \times \frac{2}{7 \sqrt{2}}$ A Step-by-Step Guide

Hey guys! Let's dive into this interesting mathematical problem: $7 \sqrt{8} \times \frac{2}{7 \sqrt{2}}$. At first glance, it might seem a bit intimidating with its square roots and fractions, but don't worry, we'll break it down step-by-step and you'll see it's actually quite manageable. Our goal here is not just to find the answer, but to understand the process and the underlying mathematical principles. Understanding these principles makes tackling similar problems in the future a breeze.

Deconstructing the Expression: A Gentle Approach

So, where do we even begin? Well, in this mathematical expression, the key is simplification. We're presented with a multiplication problem involving a combination of integers and square roots. Our initial goal is to simplify the terms individually before performing the multiplication. This approach helps to minimize errors and makes the entire process more transparent. Let’s kick things off by taking a closer look at the term $7 \sqrt{8}$. The presence of the square root of 8, or $\sqrt{8}$, immediately suggests that there might be a simpler way to express it. Remember, simplifying square roots often involves finding perfect square factors within the radicand (the number inside the square root symbol). In this case, 8 can be expressed as $4 \times 2$, where 4 is a perfect square. This is our golden ticket! By rewriting $\sqrt{8}$ as $\sqrt{4 \times 2}$, we can further simplify it.

Simplifying Square Roots: Unleashing the Potential

Now, let’s put that perfect square factor to work. We know that $\sqrt{8}$ is equivalent to $\sqrt{4 \times 2}$. A fundamental property of square roots allows us to separate the product inside the square root: $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$. Applying this property, we can rewrite $\sqrt{4 \times 2}$ as $\sqrt{4} \times \sqrt{2}$. But why is this helpful? Because we know the square root of 4! It’s simply 2. So, we now have $\sqrt{4} \times \sqrt{2} = 2 \times \sqrt{2}$, or simply $2\sqrt{2}$. Notice how we’ve transformed $\sqrt{8}$ into a much simpler form, $2\sqrt{2}$. This is a crucial step in simplifying the overall expression. Now, let's go back to our original term, $7\sqrt{8}$. We can substitute our simplified form of $\sqrt{8}$ into this term. This means $7\sqrt{8}$ becomes $7 \times (2\sqrt{2})$, which equals $14\sqrt{2}$. We've successfully simplified the first part of our expression. Take a deep breath and appreciate the progress we've made! We’ve handled the square root simplification like pros.

Tackling the Fraction: A Division Dilemma?

Okay, we've conquered the $7\sqrt{8}$ part, now let's shift our attention to the fraction $\frac{2}{7\sqrt{2}}$. This term looks a little different, but the same principles of simplification apply. We have a fraction with a square root in the denominator. While this isn't inherently wrong, it's generally considered good mathematical practice to rationalize the denominator. Rationalizing the denominator means getting rid of any square roots in the bottom part of the fraction. How do we do this? The trick is to multiply both the numerator (the top part of the fraction) and the denominator (the bottom part) by the same value, specifically the square root that's causing the problem. In this case, that's $\sqrt{2}$. Multiplying both the top and bottom of the fraction by $\sqrt{2}$ doesn't change the value of the fraction itself, because we're essentially multiplying by 1 ($\frac{\sqrt{2}}{\sqrt{2}} = 1$). So, we have $\frac{2}{7\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}}$. Let's break down what happens when we multiply this out.

Rationalizing the Denominator: Taming the Square Root

When we multiply the fractions, we multiply the numerators together and the denominators together. This gives us $\frac{2 \times \sqrt{2}}{7\sqrt{2} \times \sqrt{2}}$. Let's focus on the denominator first: $7\sqrt{2} \times \sqrt{2}$. We can rewrite this as $7 \times (\sqrt{2} \times \sqrt{2})$. Now, here's a key point: when you multiply a square root by itself, you get the number inside the square root. In other words, $\sqrt{2} \times \sqrt{2} = 2$. So, our denominator becomes $7 \times 2 = 14$. Now, let's look at the numerator: $2 \times \sqrt{2}$ is simply $2\sqrt{2}$. Putting it all together, our fraction becomes $\frac{2\sqrt{2}}{14}$. We’re not quite done yet, though. We can simplify this fraction further by noticing that both the numerator and denominator have a common factor of 2. Dividing both the top and bottom by 2, we get $\frac{\sqrt{2}}{7}$. Fantastic! We've successfully rationalized the denominator and simplified the fraction.

Putting it All Together: The Grand Finale

Alright guys, we've simplified both parts of our expression. We found that $7\sqrt{8}$ simplifies to $14\sqrt{2}$, and $\frac{2}{7\sqrt{2}}$ simplifies to $\frac{\sqrt{2}}{7}$. Now we can finally perform the multiplication: $14\sqrt{2} \times \frac{\sqrt{2}}{7}$. To multiply these terms, we multiply the integers and the square roots separately. So, we have $(14 \times \frac{1}{7}) \times (\sqrt{2} \times \sqrt{2})$. Let's tackle the integers first: $14 \times \frac{1}{7} = 2$. And we already know that $\sqrt{2} \times \sqrt{2} = 2$. Therefore, our expression simplifies to $2 \times 2$, which equals 4. Woohoo! We've reached the final answer. The solution to $7 \sqrt{8} \times \frac{2}{7 \sqrt{2}}$ is 4.

Reflecting on the Journey: Key Takeaways

So, what did we learn on this mathematical adventure? We saw the power of simplification in making complex expressions more manageable. We learned how to simplify square roots by identifying perfect square factors. We also mastered the technique of rationalizing the denominator to eliminate square roots from the bottom of fractions. But perhaps the most important takeaway is the value of breaking down a problem into smaller, more digestible steps. By tackling each part of the expression individually, we were able to navigate the challenges and arrive at the solution with confidence. Remember, mathematics is not just about finding the answer, it's about understanding the process and developing problem-solving skills that you can apply to all sorts of challenges. Keep practicing, keep exploring, and keep unlocking the mysteries of mathematics!