Smallest B Value: Calculate A + B For Real Numbers

Hey guys! Let's dive into this math problem together. It looks like we need to find the smallest possible value for 'b' given the equation $2\sqrt{27} - \sqrt{12} = a\sqrt{b}$, where 'a' and 'b' are real numbers. Then, we need to figure out what a + b equals. Sounds like a fun challenge, right? So, grab your pencils and let's get started!

Understanding the Equation

Okay, so the core of this problem is the equation $2\sqrt{27} - \sqrt{12} = a\sqrt{b}$. Our mission, should we choose to accept it (and we do!), is to simplify the left side of this equation. By simplifying the equation, we aim to express it in a form that allows us to directly identify the values of 'a' and 'b'. This is crucial because the question specifically asks for the smallest value of 'b', which hints that there might be different ways to express the equation, but only one will give us the smallest 'b'. To break it down, we have terms with square roots, and we need to make sure we simplify those square roots as much as possible. This usually involves finding perfect square factors within the numbers under the square root. Once we simplify the square roots, we can combine like terms and hopefully isolate a single term with a square root, which will then directly correspond to the $a\sqrt{b}$ form. The approach to solving this involves a combination of algebraic manipulation and number sense. We're not just blindly applying formulas; we're thinking strategically about how to rewrite the equation in the most informative way. It's like we're detectives, and the simplified equation is the key piece of evidence that will lead us to the solution. We'll be paying close attention to the coefficients and the numbers under the square roots, looking for patterns and opportunities to simplify. Let's do this!

Simplifying the Left Side: $2${sqrt{27}}$ - ${sqrt{12}}$$

Alright, let's tackle the left side of the equation: $2\sqrt{27} - \sqrt{12}$. Our mission here is to simplify these square roots as much as we can. Remember, the goal is to find perfect square factors within 27 and 12. For \sqrt{27}, we can rewrite 27 as 9 * 3. And guess what? 9 is a perfect square (3 * 3). So, \sqrt{27} becomes \sqrt{9 * 3}, which we can further simplify as \sqrt{9} * \sqrt{3}, or 3\sqrt{3}. Therefore, the term $2\sqrt{27}$ becomes 2 * 3\sqrt{3}, which is 6\sqrt{3}. Now, let's handle \sqrt{12}. We can rewrite 12 as 4 * 3, and 4 is also a perfect square (2 * 2). So, \sqrt{12} becomes \sqrt{4 * 3}, which simplifies to \sqrt{4} * \sqrt{3}, or 2\sqrt{3}. Putting it all together, the left side of the equation, $2\sqrt{27} - \sqrt{12}$, can be rewritten as 6\sqrt{3} - 2\sqrt{3}. Now, notice that we have like terms – both terms have \sqrt{3}. This means we can combine them just like we would combine 6x - 2x. So, 6\sqrt{3} - 2\sqrt{3} simplifies to (6 - 2)\sqrt{3}, which is 4\sqrt{3}. Awesome! We've significantly simplified the left side of the equation. This simplified form, 4\sqrt{3}, is much easier to work with and directly relates to the $a\sqrt{b}$ form we're aiming for. This is like finding a clear path through the jungle – we're getting closer to our destination.

Matching the Form: $a${sqrt{b}}$

Okay, now we've simplified the left side of the equation to $4\sqrt{3}$. Remember our original equation: $2\sqrt{27} - \sqrt{12} = a\sqrt{b}$. And we simplified the left side to $4\sqrt{3}$. So, we now have $4\sqrt{3} = a\sqrt{b}$. This is fantastic! We're almost there. Now, we need to match this simplified form with the $a\sqrt{b}$ format. This is actually pretty straightforward. If you look closely, you can see that 4 is in the place of 'a', and 3 is under the square root, which corresponds to 'b'. So, it seems like a = 4 and b = 3. But wait! We need to be absolutely sure that b = 3 is the smallest possible value for 'b'. This is a crucial point in the problem. Is there any other way we can rewrite $4\sqrt{3}$ in the form $a\sqrt{b}$? Let's think about this. We could potentially move the 4 inside the square root. Remember, when you move a number inside a square root, you need to square it. So, if we move the 4 inside, it becomes $4^2$, which is 16. Therefore, we could rewrite $4\sqrt{3}$ as $\sqrt{16 * 3}$, which is $\sqrt{48}$. In this case, a would be 1 (since there's no coefficient explicitly written, it's understood to be 1) and b would be 48. But remember, the question asked for the smallest value of 'b'. We found b = 3 in our first simplification, and b = 48 in this alternative form. Clearly, 3 is smaller than 48. Therefore, we can confidently conclude that the smallest possible value for 'b' is indeed 3. So, we have a = 4 and b = 3. This step highlights the importance of carefully considering all possibilities and ensuring we've found the optimal solution based on the problem's specific requirements.

Calculating a + b

Alright, we've successfully identified the values of 'a' and 'b'! We found that a = 4 and b = 3. Now, the final step is to calculate a + b. This is the easy part! Simply add the values together: 4 + 3 = 7. So, the sum of a + b is 7. We've cracked the code! We took a seemingly complex equation, simplified it step by step, and arrived at the solution. This is a great example of how breaking down a problem into smaller, manageable parts can make even the trickiest questions solvable. It's like climbing a mountain – you don't try to climb the whole thing at once; you focus on one step at a time. And just like reaching the summit of a mountain, solving a challenging math problem gives you a fantastic sense of accomplishment. Now, let's double-check our work to make sure we haven't made any silly mistakes. We started with the equation $2\sqrt{27} - \sqrt{12} = a\sqrt{b}$, simplified the left side to $4\sqrt{3}$, identified a = 4 and b = 3 as the solution that gives the smallest 'b' value, and finally calculated a + b = 7. Everything looks good! We can confidently say that the answer is 7.

Final Answer

So, after all that brain-teasing work, we've arrived at the final answer! Given the equation $2\sqrt{27} - \sqrt{12} = a\sqrt{b}$, where a and b are real numbers, and considering the smallest value of b, the sum of a + b is 7. That's it! We've conquered this math problem. Remember, the key to solving these types of problems is to break them down into smaller, more manageable steps, simplify where you can, and always double-check your work. You guys are awesome! Keep practicing, and you'll become math problem-solving ninjas in no time! This problem highlighted the importance of simplifying radical expressions, identifying like terms, and carefully considering the constraints of the problem (like finding the smallest value of 'b'). These are all valuable skills that will help you tackle a wide range of mathematical challenges. And remember, the more you practice, the more confident and proficient you'll become. So, keep those pencils sharp and keep exploring the wonderful world of math!