Hey everyone! Let's dive into this interesting exponential equation and figure out the value of b that makes it true. We've got this equation: $\left(\frac{1}{12}\right)^{-2 b} \cdot 12^{-2 b+2}=12$. It might look a little intimidating at first, but don't worry, we'll break it down step by step and make it super clear.
Understanding the Problem
Before we jump into the math, let's make sure we understand what the problem is asking. We need to find the specific value of b that, when plugged into the equation, will make both sides equal. This involves working with exponents, fractions, and a bit of algebraic manipulation. The key here is to remember the rules of exponents and how to apply them. Exponential equations like these pop up all over the place, from calculating compound interest to modeling population growth, so mastering them is a great skill to have. Trust me, once you get the hang of these, they become almost like a fun puzzle to solve!
Breaking Down the Equation
Okay, let's get our hands dirty with the equation. The first thing we notice is that we have a fraction raised to a negative exponent, $(1/12)^{-2b}$. Remember, a negative exponent means we take the reciprocal of the base. So, $\left(\frac{1}{12}\right)^{-2b}$ is the same as $12^{2b}$. This simplifies our equation quite a bit. Now we have $12^{2b} \cdot 12^{-2b+2} = 12$. See, it's already looking less scary! The next step involves using another important rule of exponents: when you multiply terms with the same base, you add their exponents. So, we're going to add the exponents $2b$ and $-2b + 2$. This will give us a new exponent for the 12 on the left side of the equation. Once we've done that, we'll have a much simpler equation to work with. We're essentially combining those two terms into one, which brings us closer to isolating b and finding our answer. It’s all about making those strategic moves to simplify and conquer!
Applying the Rules of Exponents
Now, let's add those exponents. We have $2b + (-2b + 2)$, which simplifies to just 2. So our equation now looks like this: $12^2 = 12$. Wait a minute, something seems off here. We've simplified the left side to $12^2$, which is 144, and the right side is just 12. This means we've hit a snag in our calculations, or perhaps there's a misunderstanding of the initial steps. It’s crucial to double-check our work to ensure we haven't made any errors in applying the exponent rules or simplifying the equation. Remember, even a small mistake can lead us down the wrong path. So, let's rewind a bit and carefully review each step to pinpoint where the issue might be. It’s all part of the problem-solving process, and sometimes backtracking is the best way to move forward!
Correcting the Steps
Okay, let’s backtrack and meticulously re-examine our steps. We started with $\left(\frac{1}{12}\right)^{-2 b} \cdot 12^{-2 b+2}=12$. We correctly identified that $\left(\frac{1}{12}\right)^{-2b}$ is equivalent to $12^{2b}$. So far, so good. The equation then became $12^{2b} \cdot 12^{-2b+2} = 12$. Now, here’s where we need to be extra careful. When multiplying terms with the same base, we add the exponents: $2b + (-2b + 2)$. This simplifies to $2b - 2b + 2$, which indeed equals 2. So, the left side of the equation becomes $12^2$. Our equation is now $12^2 = 12$. This is where we realized something was amiss. We ended up with 144 = 12, which is clearly not true. This indicates that the simplification process, while mathematically sound, has led us to a point where the equation doesn't hold. The issue isn’t in the algebraic manipulation itself, but in how the equation behaves overall. We need to rethink how we approach this, perhaps by looking for a different angle or checking the original equation for any subtle nuances we might have missed. Sometimes, a fresh perspective is all it takes to unlock the solution!
Rethinking the Approach
Alright, guys, let's step back and look at the big picture. We've simplified the equation correctly, but we've arrived at a contradiction: $12^2 = 12$. This tells us that there's no value of b that will make the original equation true. Think about it – we've followed all the rules of exponents and algebra, and we've ended up with a statement that's simply not possible. So, what does this mean? It means the original equation has no solution. Sometimes in math, just like in life, there isn't always a neat answer waiting for us. This is a valuable lesson in itself. It's important to recognize when an equation is unsolvable, rather than trying to force a solution that doesn't exist. In this case, the equation is designed in such a way that no matter what value we plug in for b, the two sides will never be equal. It’s like trying to fit a square peg in a round hole – it’s just not going to happen!
Concluding No Solution
So, after all our hard work, we've reached an interesting conclusion: there is no value of b that satisfies the equation $\left(\frac{1}{12}\right)^{-2 b} \cdot 12^{-2 b+2}=12$. This might seem a bit anticlimactic, but it's a perfectly valid answer. In mathematics, it's just as important to be able to identify when a problem has no solution as it is to find a solution. We've learned a lot by working through this problem. We've reinforced our understanding of exponent rules, honed our algebraic skills, and, most importantly, learned to recognize when an equation is unsolvable. This is a crucial skill that will serve us well in more advanced math and in problem-solving in general. Remember, not every equation has a solution, and that’s okay! The journey of trying to solve it is where the real learning happens. We've explored, we've simplified, and we've concluded – and that’s a win in itself!
Final Answer
In conclusion, there is no value of b for which the equation $\left(\frac{1}{12}\right)^{-2 b} \cdot 12^{-2 b+2}=12$ holds true.