Solving For B: (1/12)^(-2b) * 12^(-2b+2) = 12

Hey guys! Today, we're diving into a math problem where we need to find the value of 'b' that makes the equation ${(\frac{1}{12})^{-2b} \cdot 12^{-2b+2} = 12}$ true. This involves working with exponents and a bit of algebraic manipulation. Let’s break it down step by step so we can all understand it clearly. So let's get started and make math fun!

Understanding the Problem

At its core, this problem is about solving an exponential equation. Exponential equations are equations where the variable appears in the exponent. To solve these, we often try to get the same base on both sides of the equation so we can equate the exponents. In our case, the base we're working with is 12, which appears in different forms throughout the equation. Our mission, should we choose to accept it, is to isolate 'b'.

Before we jump into the solution, it's super important to remember a couple of key exponent rules. These rules are going to be our best friends in this math adventure, so let's make sure we know them well:

1. Negative Exponents: When you see a negative exponent, like ${x^{-n}}$, it means the same as ${\frac{1}{x^n}}$. So, a negative exponent tells us to take the reciprocal of the base and then raise it to the positive exponent.
2. Product of Powers: When you're multiplying two powers that have the same base, you can add their exponents. Mathematically, this looks like ${x^m \cdot x^n = x^{m+n}}$. This rule will help us simplify the left side of our equation by combining the terms.

With these two rules in our toolkit, we are well-equipped to tackle the problem. Remember, math is like building with Lego bricks—each rule is a brick, and we're using these bricks to construct our solution. Let’s keep these rules in mind as we move forward, and they’ll guide us to the correct answer.

Step-by-Step Solution

Okay, let’s get our hands dirty and dive into solving for 'b'. The equation we're tackling is ${(\frac{1}{12})^{-2b} \cdot 12^{-2b+2} = 12}$. Remember, our main goal here is to isolate 'b', and we're going to do that by simplifying the equation step by step.

Step 1: Rewrite the term with the negative base

First up, we've got a fraction raised to a negative exponent. No sweat! We remember our trusty negative exponent rule: ${x^{-n} = \frac{1}{x^n}}$. But in reverse! Since we have ${(\frac{1}{12})^{-2b}}$, we can rewrite this as ${12^{2b}}$. This is because taking the reciprocal of ${\frac{1}{12}}$ gives us 12, and the negative exponent flips to positive. This transformation makes our equation look a bit cleaner and easier to work with.

So now, our equation looks like this: ${12^{2b} \cdot 12^{-2b+2} = 12}$. See? Already a bit friendlier.

Step 2: Combine the terms on the left side

Next, we notice that we're multiplying two terms with the same base (12). This is where our product of powers rule comes into play: ${x^m \cdot x^n = x^{m+n}}$. We're going to add the exponents of the terms on the left side. So, we add ${2b}$ and ${(-2b + 2)}$.

Adding these exponents gives us ${2b + (-2b + 2) = 2b - 2b + 2 = 2}$. This simplifies things beautifully! Now, our left side is just ${12^2}$.

Our equation now looks like this: ${12^2 = 12}$. We’re getting closer to the finish line, guys!

Step 3: Rewrite the right side with an exponent

Now, let's turn our attention to the right side of the equation. We have simply ${12}$, but to make it comparable to the left side, we can think of it as ${12^1}$. Any number raised to the power of 1 is just the number itself.

So, our equation becomes: ${12^2 = 12^1}$. This sets us up perfectly for the final step.

Step 4: Equate the exponents (incorrect step identified)

Woah there, hold on a second! Here's where we hit a snag. Looking at our equation ${12^2 = 12^1}$, we might be tempted to equate the exponents and say ${2 = 1}$. But wait a minute... that’s not right! 2 definitely does not equal 1. This is a crucial point where we need to pause and re-evaluate.

We’ve made a mistake in our logic. The equation ${12^2 = 12^1}$ is actually telling us that there's something wrong with the original problem or that there is no solution for 'b' that would make the original equation true. Think about it: 12 squared (144) will never equal 12 to the power of 1 (which is just 12). They are fundamentally different values.

Step 5: Conclusion (Corrected)

Alright, guys, let's face the music: there's no value of 'b' that will make our original equation true. We went through the steps carefully, and we ended up with a mathematical impossibility. This doesn't mean we failed; it means we solved the problem and found out that the solution doesn't exist. Sometimes, that's the answer!

So, the final, correct answer is that there is no solution for 'b'.

Common Mistakes to Avoid

When tackling problems like this, there are a few common pitfalls that students often stumble into. Knowing these mistakes can help us steer clear of them in the future.

1. Incorrectly Applying Exponent Rules: Exponent rules are powerful tools, but they need to be used precisely. For example, confusing the product of powers rule ${x^m \cdot x^n = x^{m+n}}$ with something else can lead to errors. Always double-check which rule applies to the situation.
2. Forgetting the Negative Exponent Rule: The negative exponent rule can be a bit tricky. Remember that ${x^{-n}}$ means ${\frac{1}{x^n}}$, not rac{-1}{x^n}). Getting this wrong can flip your fractions and throw off your calculations.
3. Rushing to Equate Exponents: As we saw in our solution, it’s tempting to equate exponents as soon as we have the same base on both sides. However, we must ensure the equation is logically sound before doing this. In our case, equating exponents led to a contradiction, signaling that there was no solution.
4. Not Double-Checking the Answer: Always, always, always double-check your work! Plug your solution back into the original equation to see if it holds true. If it doesn't, you know there's an error somewhere.

Practice Makes Perfect

To really nail these types of problems, practice is key. The more you work with exponents and algebraic manipulations, the more comfortable you'll become. Here are a few practice problems you can try:

1. Solve for x: ${ (\frac{1}{5})^{-x} \cdot 5^{-x+2} = 25 }$
2. Find y if: ${ (\frac{1}{3})^{-2y} \cdot 3^{-2y+1} = 9 }$
3. Determine z: ${ (\frac{1}{2})^{-3z} \cdot 2^{-3z+3} = 8 }$

Work through these problems, and don't be afraid to make mistakes. Mistakes are learning opportunities in disguise! By identifying where you went wrong, you can strengthen your understanding and build your problem-solving skills.

Wrapping Up

So, guys, we've taken a deep dive into solving for 'b' in the equation ${(\frac{1}{12})^{-2b} \cdot 12^{-2b+2} = 12}$. We learned how to use exponent rules to simplify the equation and discovered that, in this case, there's no value of 'b' that makes the equation true. We also discussed common mistakes to avoid and the importance of practice. Remember, math is a journey, and every problem we solve makes us a little bit stronger. Keep practicing, keep asking questions, and most importantly, keep having fun with math!