Solve 3^{-2x+1} . 3^{-2x-3} = 3^{-x}

Hey math whizzes and equation explorers! Today, we're diving headfirst into the awesome world of exponential equations. These are the kinds of problems that look a little intimidating at first glance, but trust me, once you get the hang of the rules, they become super fun puzzles to solve. We're going to break down a specific problem: 3^{-2 x+1} ullet 3^{-2 x-3}=3^{-x}. Don't let those negative exponents and variables in the exponents scare you off, guys. We'll tackle this step-by-step, making sure you understand every move. Our main goal here is to find the value of 'x' that makes this entire equation true. This isn't just about getting an answer; it's about understanding the laws of exponents and how they apply to solve for unknowns. So, grab your calculators, your notebooks, and let's get ready to unravel this mathematical mystery together. We'll ensure you walk away from this feeling confident and ready to face more exponential challenges. This particular equation involves the base '3', which simplifies things significantly because we can use the properties of exponents when the bases are the same. We'll be using a key rule: when you multiply terms with the same base, you add their exponents. Keep that in mind as we move through the solution. It's the cornerstone of solving this type of problem. Remember, practice is key, and by dissecting this example thoroughly, you're building a strong foundation for all sorts of exponential equation problems you might encounter in your math journey. Let's get started!

Understanding the Core Principles: Laws of Exponents

Before we jump into solving 3^{-2 x+1} ullet 3^{-2 x-3}=3^{-x}, let's quickly refresh some fundamental laws of exponents. These rules are your best friends when dealing with powers and are absolutely crucial for simplifying and solving exponential equations. The most important rule for this problem is the product of powers rule: a^m ullet a^n = a^{m+n}. This rule states that when you multiply two exponential terms with the same base ('a' in this case), you can simply add their exponents ('m' and 'n'). Think of it like this: if you have $2^3$ (which is 2 ullet 2 ullet 2) and you multiply it by $2^2$ (which is 2 ullet 2), you end up with 2 ullet 2 ullet 2 ullet 2 ullet 2, which is $2^5$. So, $3+2=5$. Easy peasy, right? Another rule, though not directly used in the simplification step here but good to remember, is the quotient of powers rule: $a^m / a^n = a^{m-n}$. And the power of a power rule: (a^m)^n = a^{m ullet n}. For our current equation, the product of powers rule is the star of the show. The base '3' is consistent throughout the equation, which is a huge hint that we'll be using this rule extensively. The exponents themselves, $-2x+1$ and $-2x-3$, are expressions containing our variable 'x'. Our job is to combine these expressions using the exponent rules and then isolate 'x'. Understanding these basic laws is like having a secret code to unlock complex mathematical problems. Without them, exponential equations can seem like an insurmountable challenge. But with them, they transform into a series of logical steps. So, keep these rules handy, visualize them, and understand why they work. This deeper comprehension will make solving problems like the one we're about to tackle much more intuitive and less like rote memorization. The elegance of these rules lies in their consistency, applying whether the exponents are positive, negative, or even fractions.

Step-by-Step Solution: Cracking the Code

Alright, team, let's get down to business with our equation: 3^{-2 x+1} ullet 3^{-2 x-3}=3^{-x}. Our first move is to simplify the left side of the equation using the product of powers rule we just discussed. Remember, when bases are the same, we add the exponents. So, we'll combine $-2x+1$ and $-2x-3$. This gives us: $3^{(-2x+1) + (-2x-3)} = 3^{-x}$. Now, let's simplify the exponent on the left side: $(-2x+1) + (-2x-3) = -2x + 1 - 2x - 3$. Combining like terms, we get $-4x - 2$. So, our equation now looks like this: $3^{-4x - 2} = 3^{-x}$.

This is a crucial step because we now have the same base ('3') on both sides of the equation. When the bases are identical, it means the exponents must be equal for the equation to hold true. This is a fundamental property that allows us to convert an exponential equation into a linear one, which is much easier to solve. Think of it this way: if $3^A = 3^B$, then it must be that $A=B$. So, we can now set the exponents equal to each other: $-4x - 2 = -x$.

Now, we have a simple linear equation to solve for 'x'. Our goal is to get all the 'x' terms on one side and the constant terms on the other. Let's add $4x$ to both sides of the equation: $-4x - 2 + 4x = -x + 4x$. This simplifies to $-2 = 3x$. Finally, to isolate 'x', we divide both sides by 3: $-2 / 3 = 3x / 3$. And there you have it: $x = -2/3$.

So, the solution to the equation 3^{-2 x+1} ullet 3^{-2 x-3}=3^{-x} is $x = -2/3$. We went from a complex-looking exponential equation to a simple fraction by applying the laws of exponents systematically. This process demonstrates the power of breaking down problems into manageable steps and using established mathematical rules. It's all about simplification and logical progression. The key takeaway is that by mastering the properties of exponents, you can effectively handle a wide range of these equations. We've successfully navigated the complexities, and hopefully, you guys feel a bit more comfortable with these types of problems now!

Verifying the Solution: Does it Hold Up?

So, we found our potential answer: $x = -2/3$. But in the world of mathematics, especially when dealing with equations, it's always a smart move to verify our solution. This means plugging our value of 'x' back into the original equation to see if it makes both sides equal. It's like double-checking your work to ensure you haven't made any silly mistakes. This verification step is super important because it confirms that our answer is indeed correct and that we truly understand the concept.

Our original equation is 3^{-2 x+1} ullet 3^{-2 x-3}=3^{-x}. Let's substitute $x = -2/3$ into the left side first:

• Exponent 1: $-2x + 1 = -2(-2/3) + 1 = 4/3 + 1 = 4/3 + 3/3 = 7/3$
• Exponent 2: $-2x - 3 = -2(-2/3) - 3 = 4/3 - 3 = 4/3 - 9/3 = -5/3$

So, the left side becomes 3^{7/3} ullet 3^{-5/3}. Using the product of powers rule (a^m ullet a^n = a^{m+n}), we add the exponents:

$7/3 + (-5/3) = 7/3 - 5/3 = 2/3$.

Therefore, the left side simplifies to $3^{2/3}$.

Now, let's evaluate the right side of the original equation with $x = -2/3$:

• Exponent: $-x = -(-2/3) = 2/3$.

So, the right side is $3^{2/3}$.

Comparing the simplified left side ($3^{2/3}$) and the right side ($3^{2/3}$), we see that they are indeed equal! $3^{2/3} = 3^{2/3}$. This confirms that our solution $x = -2/3$ is correct. This verification process is not just a formality; it reinforces our understanding of how the properties of exponents work and builds confidence in our problem-solving abilities. It's a critical step in ensuring mathematical accuracy and demonstrating mastery of the subject. So, next time you solve an equation, don't forget to check your answer – it's a habit that pays off big time!

Conclusion: Mastering Exponential Equations

So there you have it, math explorers! We've successfully tackled the exponential equation 3^{-2 x+1} ullet 3^{-2 x-3}=3^{-x} and found that $x = -2/3$. We did this by leaning on the fundamental laws of exponents, particularly the product of powers rule, which allowed us to combine terms with the same base by adding their exponents. This simplification led us to an equation where we could equate the exponents directly: $-4x - 2 = -x$. Solving this linear equation was the final step to uncovering our value for 'x'.

What's the big takeaway here, guys? It's that seemingly complex problems can be demystified by breaking them down into smaller, manageable steps and applying the correct mathematical principles. Understanding the 'why' behind the rules, like why a^m ullet a^n = a^{m+n}, makes the process much more intuitive. We also performed a crucial verification step, plugging our solution back into the original equation to confirm its accuracy. This practice is invaluable for building confidence and ensuring correctness in your mathematical endeavors.

Exponential equations are a fundamental part of algebra and appear in various fields, from finance to science. By mastering the techniques we used today – understanding exponent properties, simplifying expressions, and solving resulting equations – you're building a strong foundation. Keep practicing, keep exploring, and don't be afraid to tackle new challenges. The world of mathematics is vast and rewarding, and every problem you solve opens up new possibilities. Remember, the key is consistent effort and a willingness to engage with the material. You've got this!