Proving A Number Is Rational: A Step-by-Step Guide
Hey guys! Today, we're diving into a super interesting math problem where we need to show that a given number, let's call it 'a', is rational. Now, before we jump into the nitty-gritty details, let's quickly recap what a rational number actually is. Simply put, a rational number is any number that can be expressed as a fraction p/q, where p and q are integers (whole numbers) and q is not zero. So, our mission today is to manipulate this somewhat intimidating expression for 'a' and see if we can massage it into a nice, neat fraction. Ready to get started?
Understanding the Problem
The expression we're dealing with is: a = 3√((4√27)/ (√3 · 3√2)) · (9√16)/(3 12√3). At first glance, it looks like a jumbled mess of radicals and numbers, right? But don't worry, we're going to break it down step by step. The key here is to remember the properties of radicals and exponents. We'll be using these properties to simplify the expression and, hopefully, reveal its rational nature. Before we get lost in the complexity, let's highlight the main goal: to demonstrate that 'a' can be written as a fraction. This involves simplifying the radicals, combining terms, and ultimately expressing the entire number in the form of p/q.
Breaking Down the Expression
So, where do we even start with something like this? A good approach is to tackle it piece by piece. Let's first focus on the terms inside the cube root and then move outwards. We'll simplify the square roots and cube roots individually, and then see how they interact with each other. Remember, the goal is to eliminate the radicals if possible and combine like terms. This part is all about careful manipulation and keeping track of every step. We want to avoid mistakes, so double-checking our work as we go is a great idea. Think of it like untangling a knot – slow and steady wins the race!
Simplifying Radicals
The heart of this problem lies in simplifying radicals. Guys, remember that radicals are just another way of expressing exponents. For example, √x is the same as x^(1/2), and ∛x is the same as x^(1/3). This connection between radicals and exponents is crucial for simplifying expressions. When we see a radical, we should immediately think about whether we can rewrite the number inside it as a power. This often helps in simplifying the entire expression. Let's start by looking at the square roots and cube roots in our expression and see what we can simplify.
Working with Square Roots
Let's focus on the square roots first. We have √27, √3, and √16 in our expression. Can we simplify these? Absolutely! Remember, √27 can be written as √(9 * 3), which simplifies to 3√3. And √16 is a perfect square – it's simply 4. √3 is already in its simplest form, so we'll leave it as is. By simplifying these square roots, we're already making progress towards a cleaner expression. This step-by-step approach is key to avoiding confusion and making the problem more manageable. It’s like chopping up a big task into smaller, more digestible chunks.
Tackling Cube Roots
Now, let's turn our attention to the cube roots. We have 3√((4√27)/ (√3 · 3√2)) and 12√3 in our expression. Simplifying cube roots is similar to simplifying square roots, but this time we're looking for perfect cubes. The key here is to rewrite the numbers inside the cube roots as powers of 3. This might involve some manipulation and clever rearranging of terms. Don't be afraid to experiment and try different approaches. Math is all about exploring and finding the most efficient path to the solution. We're essentially playing a puzzle, and the pieces are these radical expressions.
Combining Terms and Exponents
After simplifying the radicals, the next step is to combine like terms and use the properties of exponents to further simplify the expression. Remember, when multiplying terms with the same base, we add the exponents. And when dividing, we subtract the exponents. These rules are fundamental to simplifying expressions with exponents and radicals. We'll be using them extensively in this section.
Using Exponent Rules
Now, let’s put those exponent rules into action. Our goal is to rewrite the entire expression in terms of powers. This will allow us to easily combine terms and see if we can eliminate the radicals altogether. This step often involves converting radicals back into exponents and then applying the rules of exponents. It's like translating from one language to another – we're taking the expression from radical form to exponential form. And once we're in exponential form, the simplification process becomes much smoother. This is where the magic of math really starts to shine!
Simplifying the Fraction
Our expression probably involves fractions within fractions at this point. Simplifying these complex fractions is crucial. Remember, dividing by a fraction is the same as multiplying by its reciprocal. We'll be using this principle to get rid of the inner fractions and consolidate the expression. Think of it like cleaning up a messy room – we're getting rid of the clutter and organizing everything in a neat and tidy way. A well-organized expression is much easier to work with and allows us to see the final result more clearly. This meticulous approach is what distinguishes good mathematicians from the rest.
Proving Rationality
Finally, we're at the last stage! After all the simplification, we should have an expression that looks much simpler than the original. Our goal now is to show that this simplified expression is a rational number. This means we need to express it in the form p/q, where p and q are integers and q is not zero. If we can do this, we've successfully proven that the number 'a' is rational. This is the grand finale of our mathematical journey!
Expressing as p/q
Guys, if we've done everything correctly, the simplified expression should either be an integer or a simple fraction. If it's an integer, we can easily write it as p/1, which fits the definition of a rational number. If it's a fraction, we just need to make sure that both the numerator and denominator are integers. This final step is like the last piece of a puzzle falling into place. It's the moment of triumph when we see that all our hard work has paid off.
Conclusion
So, there you have it! We've successfully navigated the complex world of radicals and exponents, simplified a seemingly intimidating expression, and proven that the number 'a' is rational. This problem highlights the power of step-by-step problem-solving and the importance of understanding fundamental mathematical principles. Remember, math isn't about memorizing formulas; it's about understanding the underlying concepts and applying them creatively. Keep practicing, keep exploring, and most importantly, keep having fun with math! You got this! This exercise not only sharpens our skills in simplifying radical expressions but also reinforces the concept of rational numbers. By meticulously breaking down the problem and applying the rules of exponents and radicals, we can transform complex expressions into simpler, manageable forms, ultimately revealing their true nature. This process is a testament to the elegance and power of mathematical reasoning. You've done great! Keep up the amazing work, and remember that every complex problem is just a series of simpler steps waiting to be solved.