Solving: 3^(x-2) * 9^x = (1/3^x) * (3^(1-x))^2

Hey guys! Today, we are diving into an exciting math problem involving exponential equations. We're going to break down the steps to solve the equation 3^(x-2) * 9^x = (1/3^x) * (3^(1-x))2. Exponential equations might seem intimidating at first, but don't worry! We'll tackle it together, step by step. Understanding how to solve these types of equations is super useful in various fields, from science and engineering to finance. So, let's get started and unravel this mathematical puzzle!

Understanding the Basics of Exponential Equations

Before we jump into solving the specific equation, let's quickly review the fundamentals of exponential equations. In essence, an exponential equation is one where the variable appears in the exponent. Key to solving these equations is understanding and applying the properties of exponents. These properties allow us to manipulate the equations, simplify them, and ultimately isolate the variable. For example, we frequently use the rule that a^(m) * a^(n) = a^(m+n) to combine terms with the same base. Another crucial property is (a^m)n = a^(mn)*, which helps us deal with powers raised to powers. Also, remember that a^(-n) = 1/a^n, which is useful for handling reciprocals. Mastering these rules is essential for solving more complex equations efficiently. Understanding these basics will make the process much smoother and less daunting. So, keep these rules handy as we move forward!

Key Properties of Exponents to Remember

When dealing with exponential equations, several key properties of exponents come into play. These properties are the tools we use to simplify and solve the equations. Let's go over some of the most important ones:

1. Product of Powers: a^(m) * a^(n) = a^(m+n). This rule states that when you multiply powers with the same base, you can add the exponents.
2. Quotient of Powers: a^(m) / a^(n) = a^(m-n). When you divide powers with the same base, you subtract the exponents.
3. Power of a Power: (a^m)n = a^(mn)*. If you have a power raised to another power, you multiply the exponents.
4. Power of a Product: (ab)^n = a^n * b^n. This property allows you to distribute an exponent over a product.
5. Power of a Quotient: (a/b)^n = a^n / b^n. Similar to the power of a product, this distributes an exponent over a quotient.
6. Negative Exponent: a^(-n) = 1/a^n. A negative exponent indicates a reciprocal.
7. Zero Exponent: a^0 = 1 (if a ≠ 0). Any non-zero number raised to the power of 0 is 1.

These properties are the foundation for manipulating and solving exponential equations. By applying these rules correctly, we can simplify complex expressions and make the equations easier to handle. Make sure you're comfortable with these properties, as we'll be using them extensively in our solution.

Step-by-Step Solution to 3^(x-2) * 9^x = (1/3^x) * (3^(1-x))2

Alright, let's dive into solving the equation 3^(x-2) * 9^x = (1/3^x) * (3^(1-x))2. We're going to take it one step at a time, so it’s super clear and easy to follow. Remember, the key to these problems is to simplify and consolidate using the properties of exponents we talked about earlier. So, keep those rules in mind as we work through each step together. By the end, you’ll not only have the solution but also a solid understanding of how we got there. Let’s get started and break this down!

Step 1: Express all terms with the same base

The first crucial step in solving this exponential equation is to express all terms with the same base. Looking at our equation, 3^(x-2) * 9^x = (1/3^x) * (3^(1-x))2, we notice that we have bases of 3 and 9. Since 9 is a power of 3 (specifically, 9 = 3^2), we can rewrite the equation using 3 as the common base. This is a critical move because it allows us to combine terms more easily later on. So, let's replace 9 with 3^2 in our equation. This gives us: 3^(x-2) * (3²⁾x = (1/3^x) * (3^(1-x))2. Now, everything is in terms of base 3, which sets us up nicely for the next step. Remember, the goal here is to make the equation as uniform as possible, and having a common base is a huge part of that. Keep this strategy in mind for other exponential equations you encounter!

Step 2: Simplify using exponent rules

Now that we have a common base, it's time to simplify the equation using the properties of exponents. Our equation currently looks like this: 3^(x-2) * (3²⁾x = (1/3^x) * (3^(1-x))2. Let's tackle the terms one by one. First, we have (3²⁾x. Using the power of a power rule, which states that (a^m)n = a^(mn), we can simplify this to 3^(2x). Next, let's look at the term (1/3^x). Recall that a negative exponent indicates a reciprocal, so we can rewrite this as 3^(-x). Finally, we have (3^(1-x))2. Again, using the power of a power rule, we multiply the exponents to get 3^(2(1-x)) = 3^(2-2x). Substituting these simplifications back into our equation, we now have: 3^(x-2) * 3^(2x) = 3^(-x) * 3^(2-2x). See how much cleaner the equation looks now? This simplification is key to making the next steps manageable. We're on the right track!

Step 3: Combine terms on each side of the equation

With our simplified equation 3^(x-2) * 3^(2x) = 3^(-x) * 3^(2-2x), the next step is to combine the terms on each side. We'll use the product of powers rule, which states that a^(m) * a^(n) = a^(m+n). On the left side, we have 3^(x-2) * 3^(2x). Adding the exponents, we get 3^((x-2) + 2x) = 3^(3x-2). On the right side, we have 3^(-x) * 3^(2-2x). Adding these exponents gives us 3^((-x) + (2-2x)) = 3^(2-3x). So, our equation now simplifies to 3^(3x-2) = 3^(2-3x). Notice how combining like terms has made our equation even more straightforward. We're getting closer to isolating x and finding our solution. Keep pushing forward!

Step 4: Set the exponents equal to each other

Now that we have the equation in the form 3^(3x-2) = 3^(2-3x), we can take a significant step towards solving for x. Since the bases are the same (both are 3), the exponents must be equal for the equation to hold true. This means we can set the exponents equal to each other: 3x - 2 = 2 - 3x. This is a crucial step because it transforms our exponential equation into a simple linear equation, which is much easier to solve. We've effectively eliminated the exponential part and are now dealing with a basic algebraic problem. This is a common strategy in solving exponential equations: once you have the same base on both sides, you can equate the exponents. Now, let's solve this linear equation and find the value of x!

Step 5: Solve for x

We've arrived at the linear equation 3x - 2 = 2 - 3x. Now, let's solve for x. First, we want to get all the x terms on one side and the constants on the other. Add 3x to both sides of the equation: 3x - 2 + 3x = 2 - 3x + 3x, which simplifies to 6x - 2 = 2. Next, add 2 to both sides: 6x - 2 + 2 = 2 + 2, which simplifies to 6x = 4. Finally, divide both sides by 6 to isolate x: 6x / 6 = 4 / 6, which gives us x = 4/6. We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2. So, x = 4/6 simplifies to x = 2/3. And there we have it! We've solved for x. Remember, it's always a good idea to check your solution by plugging it back into the original equation to make sure it holds true. Great job!

Final Answer: x = 2/3

So, after carefully working through each step, we've found that the solution to the exponential equation 3^(x-2) * 9^x = (1/3^x) * (3^(1-x))2 is x = 2/3. It’s always satisfying to reach the final answer, right? But more than that, it’s super important to understand the process we used to get there. We started by making sure all the terms had the same base, then simplified using exponent rules, combined like terms, and finally solved for x. Each step built upon the previous one, and that’s often how it goes with math problems. If you encounter a similar problem in the future, try to apply these same strategies. Remember to take it one step at a time, and don't be afraid to revisit the exponent rules we discussed. You've got this!

Verification of the Solution

To ensure our solution x = 2/3 is correct, we should substitute it back into the original equation: 3^(x-2) * 9^x = (1/3^x) * (3^(1-x))2. Let's plug in x = 2/3 and see if both sides of the equation are equal. On the left side, we have 3^((2/3)-2) * 9^(2/3). This simplifies to 3^(-4/3) * (3²⁾(2/3) = 3^(-4/3) * 3^(4/3). When we multiply these terms, we add the exponents: 3^(-4/3 + 4/3) = 3^0 = 1. Now, let's look at the right side of the equation: (1/3^(2/3)) * (3^(1-(2/3)))2. This simplifies to 3^(-2/3) * (3^(1/3))2 = 3^(-2/3) * 3^(2/3). Adding the exponents, we get 3^(-2/3 + 2/3) = 3^0 = 1. Since both sides of the equation equal 1 when we substitute x = 2/3, our solution is verified. This step is crucial because it confirms that we haven't made any errors in our calculations and that our answer is indeed correct. Always remember to verify your solutions whenever possible!

Tips and Tricks for Solving Exponential Equations

Solving exponential equations can become much easier with a few handy tips and tricks up your sleeve. Let's go over some strategies that can help you tackle these problems more efficiently. First off, always try to express all terms with the same base. This is often the key to unlocking the solution, as it allows you to equate exponents, as we did in our example. Another tip is to simplify the equation as much as possible before diving into more complex steps. Use exponent rules to combine terms and clear out any unnecessary clutter. Also, keep an eye out for opportunities to substitute variables. If you notice a repeating exponential expression, substituting a single variable for that expression can simplify the equation significantly. Lastly, don't forget to check your solution by plugging it back into the original equation. This will help you catch any mistakes and ensure that your answer is correct. With these tips in mind, you'll be well-equipped to solve a wide range of exponential equations. Keep practicing, and you'll become a pro in no time!

Common Mistakes to Avoid

When solving exponential equations, there are a few common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure you get the correct answer. One frequent mistake is incorrectly applying the exponent rules. For example, confusing the product of powers rule with the power of a power rule can lead to errors. Make sure you have a solid grasp of these rules and know when to apply each one. Another common mistake is forgetting to distribute exponents correctly. When you have an expression like (ab)^n, remember that the exponent n applies to both a and b. Similarly, be careful when dealing with negative exponents and reciprocals. A negative exponent means you should take the reciprocal of the base, but the sign of the base itself remains the same. Finally, always double-check your work and verify your solution. It’s easy to make a small arithmetic error, so plugging your answer back into the original equation can help you catch mistakes. By being mindful of these common pitfalls, you can increase your accuracy and confidence in solving exponential equations.

Practice Problems

To really nail down your understanding of solving exponential equations, practice is key! Working through a variety of problems will help you become more comfortable with the different strategies and techniques involved. Here are a few practice problems you can try:

1. Solve for x: 2^(3x-1) = 8
2. Solve for y: 5^(2y) = 25^(y+1)
3. Solve for z: 4^(z+2) = 16^(z)
4. Solve for a: 9^(a-1) = 3^(2a)
5. Solve for b: (1/2)^(b+1) = 4^(-b)

Try tackling these problems using the steps and tips we've discussed. Remember to simplify, express terms with the same base, and check your solutions. If you get stuck, revisit the earlier sections of this guide for a refresher. The more you practice, the more confident you'll become in solving exponential equations. Good luck, and happy problem-solving!

Conclusion

We've reached the end of our journey through solving the exponential equation 3^(x-2) * 9^x = (1/3^x) * (3^(1-x))2! We took it step by step, from understanding the basics of exponential equations to applying exponent rules, simplifying, and finally, solving for x. Remember, the key to these problems is to break them down into manageable steps and to use the properties of exponents to your advantage. We also talked about some common mistakes to avoid and shared tips and tricks to make the process smoother. Most importantly, we emphasized the importance of practice. The more you work with these types of equations, the more comfortable and confident you'll become. So, keep practicing, keep exploring, and keep pushing your mathematical boundaries. You've got the tools you need to succeed!