Dividing √27: What Number Gives 18?

Hey guys! Today, we're diving into a fun little math problem that involves square roots and division. Specifically, we're tackling the question: If we divide the square root of 27 by what number will the result be 18? This might sound a bit tricky at first, but don't worry, we'll break it down step by step so it's super easy to understand. So, grab your thinking caps, and let's get started!

Understanding the Problem

Before we jump into solving the equation, let's make sure we fully understand what the problem is asking. The core question is: What number, when used as a divisor for the square root of 27, yields a quotient of 18? Think of it like this: we have √27, and we need to find a mystery number that, when we divide √27 by it, we get 18. This involves a good grasp of square roots and how they interact with division. We'll need to simplify the square root and then use some algebraic manipulation to isolate our mystery number. This kind of problem often pops up in basic algebra and geometry, so mastering it is a great step in building your math skills. It’s not just about finding the answer; it's also about understanding the process and logic involved. This way, you’ll be able to tackle similar problems with confidence. Remember, math is like building blocks, and each concept you learn helps you understand the next one better.

Breaking Down the Square Root of 27

The first key step in solving our problem is to simplify the square root of 27 (√27). Simplifying square roots makes the numbers easier to work with and helps us see the problem more clearly. So, how do we do it? Well, we need to find the largest perfect square that divides evenly into 27. A perfect square is a number that is the result of squaring a whole number (like 4, which is 2 squared, or 9, which is 3 squared). In the case of 27, the largest perfect square that divides into it is 9 because 9 multiplied by 3 equals 27. Now, we can rewrite √27 as √(9 * 3). Using the property of square roots that says √(a * b) = √a * √b, we can further break this down into √9 * √3. We all know that the square root of 9 is 3 (since 3 * 3 = 9), so we can replace √9 with 3. This leaves us with 3√3. So, we've successfully simplified √27 to 3√3. This is a crucial step because it transforms the original square root into a more manageable form for our calculations. By simplifying the square root first, we make the subsequent steps much easier and reduce the chances of making mistakes along the way. Plus, it’s just good practice to get comfortable with simplifying radicals, as it's a skill that comes up a lot in math and science!

Setting Up the Equation

Now that we've simplified √27 to 3√3, we can set up the equation to solve for our mystery number. Remember, the question is: What number, when used as a divisor for the square root of 27, yields a quotient of 18? This can be translated into a mathematical equation. Let's call our mystery number 'x'. So, we are saying that when we divide 3√3 by x, we should get 18. In equation form, this looks like: (3√3) / x = 18. Setting up the equation correctly is super important because it forms the foundation for solving the problem. If we get the equation wrong, the answer we get won't be correct either. It's like building a house; if the foundation isn't solid, the whole structure will be shaky. This equation tells us exactly what we need to do: we need to isolate 'x' on one side of the equation to find its value. To do this, we'll use some basic algebraic principles. The goal is to get 'x' by itself so we can see what number it represents. So, we've got our simplified square root and our equation ready to go. The next step is to actually solve the equation for 'x', which we'll tackle in the next section.

Solving for the Unknown

Alright, let's get down to the nitty-gritty and solve for our unknown, 'x'! We've already set up our equation as (3√3) / x = 18. Now, we need to isolate 'x' on one side of the equation. This involves a bit of algebraic maneuvering, but don't worry, it's totally manageable. The first thing we can do is multiply both sides of the equation by 'x'. This gets 'x' out of the denominator on the left side, making it easier to work with. When we do this, our equation becomes: 3√3 = 18x. See? We're getting closer! Now, 'x' is multiplied by 18, so to get 'x' by itself, we need to do the opposite operation: we'll divide both sides of the equation by 18. This will isolate 'x' on the right side. When we divide both sides by 18, we get: (3√3) / 18 = x. Great! We've got 'x' by itself. However, we're not quite done yet. We can simplify the left side of the equation to make our answer cleaner. We can simplify the fraction 3/18 by dividing both the numerator and the denominator by their greatest common divisor, which is 3. This gives us 1/6. So, our equation now looks like: (√3) / 6 = x. And that's it! We've solved for 'x'. Our mystery number is (√3) / 6.

Simplifying the Answer

So, we've found that x = (√3) / 6. But let's take a moment to appreciate what we've accomplished and ensure our answer makes sense. We started with the question: If we divide the square root of 27 by what number will the result be 18? And through simplification and algebraic manipulation, we've arrived at the answer (√3) / 6. Now, let’s think about this. The square root of 3 is roughly 1.732. When we divide that by 6, we get a number that’s approximately 0.2887. This means that if we divide √27 (which we simplified to 3√3) by approximately 0.2887, we should indeed get 18. This kind of reality check is crucial in math. It's not just about crunching numbers; it's about understanding what the numbers mean and whether our answer is logical in the context of the problem. Always ask yourself,