Simplifying Square Roots How To Simplify -√(72)

Understanding the Basics of Simplifying Square Roots

Alright, let's dive into simplifying the expression $-\sqrt{72}$. Before we get to the nitty-gritty, it's crucial to understand what we're actually doing when we simplify a square root. Essentially, we're trying to find the largest perfect square that divides evenly into the number under the square root (the radicand). A perfect square is a number that can be obtained by squaring an integer (e.g., 4, 9, 16, 25, and so on). Our goal is to rewrite the radicand as a product of a perfect square and another factor. This allows us to take the square root of the perfect square part, pulling it out from under the radical, and leaving the remaining factor inside. This process not only simplifies the expression but also gives us a clearer understanding of the number's components. When dealing with negative signs outside the square root, like in our example $-\sqrt{72}$, it's essential to keep that negative sign intact throughout the simplification process. This negative sign simply means that the final simplified expression will be the negative of the simplified square root. For example, if we simplify $\sqrt{72}$ to $6\sqrt{2}$, then $-\sqrt{72}$ would be $-6\sqrt{2}$. This is a critical detail to avoid errors and ensure accuracy in your calculations. Think of the negative sign as a coefficient of -1 multiplying the square root; it's along for the ride! Keeping track of the negative sign is often a simple step to overlook, but it makes a big difference in the final answer. So, always double-check that you've correctly carried it through every step.

Breaking Down 72 to Find Perfect Square Factors

Now, let's get specific and break down the number 72 to find its perfect square factors. This is the core of simplifying square roots, guys. The trick here is to identify the largest perfect square that divides 72. One way to do this is to start listing out the perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, and so on. Then, check which of these divides 72 evenly. You'll quickly notice that 36 is the largest perfect square that goes into 72. In fact, 72 can be written as $36 \times 2$. This is exactly what we need! By identifying 36 as the perfect square factor, we set ourselves up for easy simplification. If you didn't spot 36 right away, no worries! You can also break down 72 into smaller factors and build up to the perfect square. For example, you might initially see that 72 is $9 \times 8$. Then, you can recognize that 9 is a perfect square, and 8 can be further broken down into $4 \times 2$, where 4 is also a perfect square. Combining these, you get $9 \times 4 \times 2$, which leads you to the same conclusion – that 36 (which is $9 \times 4$) is the largest perfect square factor. This method of breaking down into smaller factors is super helpful when dealing with larger numbers or when the perfect square isn't immediately obvious. It's like detective work, piecing together the clues to reveal the hidden perfect square. Remember, the goal is to find that perfect square, because once you do, you're one step closer to simplifying the entire expression. Thinking through different factor pairs and identifying the perfect squares within them is a valuable skill that becomes more intuitive with practice. Keep at it, and you'll be spotting those perfect squares like a pro in no time!

Applying the Product Property of Square Roots

Okay, so we've broken down 72 into $36 \times 2$. Now comes the fun part: applying the product property of square roots. What is this property, you ask? Well, it's a fundamental rule that allows us to separate the square root of a product into the product of square roots. In mathematical terms, it states that $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$, where a and b are non-negative numbers. This property is a game-changer when simplifying square roots because it lets us deal with the perfect square factor separately. In our case, we have $-\sqrt{72}$, which we've rewritten as $-\sqrt{36 \times 2}$. Using the product property, we can now split this into $-\sqrt{36} \times \sqrt{2}$. See how we've isolated the perfect square (36) under its own square root? This is where the magic happens. We know that the square root of 36 is 6, so $\sqrt{36}$ simplifies to 6. The remaining factor, $\sqrt{2}$, cannot be simplified further because 2 has no perfect square factors other than 1. So, we just leave it as $\sqrt{2}$. By applying the product property, we've transformed a complex square root into a simpler form. It's like taking a complicated puzzle and breaking it into smaller, more manageable pieces. This property is not just a shortcut; it's a powerful tool for understanding the structure of square roots and how they interact with multiplication. Remember, the key is to identify the perfect square factors and then use this property to separate them out. Once you've mastered this technique, you'll be simplifying square roots with confidence and ease.

Simplifying the Square Root and Final Answer

Alright, let's bring it all together and get to the final answer. We've got $-\sqrt{72}$, which we broke down into $-\sqrt{36 \times 2}$. Then, using the product property of square roots, we separated it into $-\sqrt{36} \times \sqrt{2}$. We know that $\sqrt{36}$ is 6, so we can substitute that in. This gives us $-6 \times \sqrt{2}$, or simply $-6\sqrt{2}$. And there you have it! The simplified form of $-\sqrt{72}$ is $-6\sqrt{2}$. Notice how we kept the negative sign throughout the entire process. That's super important! The negative sign just tags along for the ride and applies to the entire simplified expression. This final answer, $-6\sqrt{2}$, is in its simplest form because the number under the square root (the radicand, which is 2) has no perfect square factors other than 1. We've successfully extracted the largest perfect square factor from the original number and simplified the expression as much as possible. This is the goal of simplifying square roots: to express them in their most basic form, making them easier to understand and work with. Simplifying square roots is like cleaning up a messy room. You start with a jumble of stuff, but by identifying the key components and organizing them, you end up with something neat and tidy. In this case, we took a seemingly complex square root and transformed it into a clean, simplified expression. This process not only gives us the answer but also helps us appreciate the underlying mathematical structure. So, next time you see a square root, remember to break it down, find those perfect squares, and simplify with confidence!

Therefore, $-\sqrt{72} = -6\sqrt{2}$.