Solving Exponential Equation (1/3)^(3x+6) = 9^(x-2) A Step-by-Step Guide

Are you grappling with exponential equations? Do you find yourself staring at exponents and bases, unsure how to proceed? Fear not! This comprehensive guide will walk you through the process of solving the equation (1/3)^(3x+6) = 9^(x-2), providing a step-by-step solution along with clear explanations of the underlying concepts. Mastering the techniques for solving exponential equations is crucial not only for academic success in mathematics but also for various applications in fields like finance, physics, and computer science. This article aims to equip you with the necessary knowledge and skills to confidently tackle a wide range of exponential problems. Let's delve into the world of exponents and unravel the mysteries behind solving these equations. From understanding the fundamental properties of exponents to applying strategic algebraic manipulations, we will explore every facet of the problem at hand. By the end of this guide, you'll not only know the solution to this particular equation but also have a solid foundation for solving similar problems in the future. So, grab your pen and paper, and let's embark on this mathematical journey together! Remember, practice makes perfect, and with consistent effort, you'll become a pro at solving exponential equations. Let's begin by understanding the core principles that govern these equations.

Understanding the Basics of Exponential Equations

Before diving into the specifics of the equation (1/3)^(3x+6) = 9^(x-2), it's essential to grasp the fundamental concepts of exponential equations. An exponential equation is an equation in which the variable appears in an exponent. These equations are characterized by a constant base raised to a variable power. The key to solving exponential equations lies in manipulating them so that both sides have the same base. This allows us to equate the exponents and solve for the variable. Understanding the properties of exponents is crucial in this process. Some of the most important properties include the product of powers rule (a^m * a^n = a^(m+n)), the quotient of powers rule (a^m / a^n = a^(m-n)), the power of a power rule ((a^m)n = a^(mn)), and the negative exponent rule (a^(-n) = 1/a^n). These properties provide the tools we need to simplify and rewrite exponential expressions. In addition to these rules, it's also important to remember that any non-zero number raised to the power of 0 is equal to 1 (a^0 = 1). This can be a useful property in certain situations. Furthermore, understanding the relationship between exponential and logarithmic functions is beneficial for solving more complex exponential equations. Logarithms are the inverse of exponential functions, and they provide an alternative way to isolate the variable when it's in the exponent. However, for this particular equation, we can solve it effectively using the properties of exponents alone. By mastering these basic concepts and properties, you'll be well-prepared to tackle a wide range of exponential equations, from simple to complex. Now, let's move on to the specific steps involved in solving our equation.

Step-by-Step Solution of (1/3)^(3x+6) = 9^(x-2)

Let's break down the solution to the equation (1/3)^(3x+6) = 9^(x-2) into manageable steps. Our primary goal is to express both sides of the equation with the same base. This will allow us to equate the exponents and solve for x.

Step 1: Express both bases as powers of the same number.

Observe that both 1/3 and 9 can be expressed as powers of 3. We can rewrite 1/3 as 3^(-1) and 9 as 3^2. Substituting these into the original equation, we get:

(3^(-1))(3x+6) = (3²⁾(x-2)

This step is crucial because it sets the stage for applying the power of a power rule. By expressing both bases as powers of the same number, we create a common ground for comparison and simplification. This transformation is a fundamental technique in solving exponential equations, allowing us to move from different bases to a unified representation. The ability to recognize and apply this step is a key skill in mastering exponential equations.

Step 2: Apply the power of a power rule.

The power of a power rule states that (a^m)n = a^(mn). Applying this rule to both sides of the equation, we have:

3^(-1*(3x+6)) = 3^(2*(x-2))

Simplifying the exponents, we get:

3^(-3x-6) = 3^(2x-4)

The power of a power rule is a cornerstone in manipulating exponential expressions. It allows us to simplify nested exponents, making the equation more manageable. This step is not just about applying a formula; it's about strategically transforming the equation to a form where we can directly compare the exponents. The application of this rule is a testament to the elegance and power of exponential properties. By understanding and utilizing this rule, we can effectively navigate through complex exponential equations.

Step 3: Equate the exponents.

Since the bases are now the same (both are 3), we can equate the exponents:

-3x - 6 = 2x - 4

This is a critical step in solving exponential equations. Once we have the same base on both sides, we can eliminate the exponential part and focus solely on the exponents. This transforms the problem from an exponential equation to a linear equation, which is much easier to solve. Equating the exponents is a direct consequence of the one-to-one property of exponential functions, which states that if a^m = a^n, then m = n. This property is a fundamental principle in solving exponential equations, and it allows us to bridge the gap between exponential expressions and linear equations.

Step 4: Solve the linear equation.

Now we have a simple linear equation. Let's solve for x:

Add 3x to both sides:

-6 = 5x - 4

Add 4 to both sides:

-2 = 5x

Divide both sides by 5:

x = -2/5

Solving linear equations is a fundamental skill in algebra, and it plays a crucial role in solving exponential equations. The steps involved in solving a linear equation are straightforward: isolate the variable by performing the same operations on both sides of the equation. In this case, we used addition and division to isolate x. The ability to confidently solve linear equations is essential for tackling more complex mathematical problems, including exponential equations. This step demonstrates the interconnectedness of different mathematical concepts, highlighting how basic algebraic skills are necessary for solving advanced problems.

Step 5: Verify the solution (Optional but Recommended).

To ensure our solution is correct, we can substitute x = -2/5 back into the original equation:

(1/3)^(3*(-2/5)+6) = 9^(-2/5-2)

Simplifying both sides:

(1/3)^(-6/5+6) = 9^(-12/5)

(1/3)^(24/5) = 9^(-12/5)

(3^(-1))(24/5) = (3²⁾(-12/5)

3^(-24/5) = 3^(-24/5)

Since both sides are equal, our solution x = -2/5 is correct.

Verifying the solution is a crucial step in the problem-solving process. It provides a check on our work and ensures that the solution we obtained is indeed correct. This step is particularly important in exponential equations, where errors can easily occur due to the manipulation of exponents. By substituting the solution back into the original equation, we can confirm that both sides of the equation are equal, thereby validating our solution. This step not only ensures accuracy but also reinforces our understanding of the problem and the solution process. It's a testament to the importance of thoroughness and attention to detail in mathematics.

Common Mistakes to Avoid

When solving exponential equations, several common mistakes can lead to incorrect answers. Being aware of these pitfalls can help you avoid them and improve your problem-solving accuracy.

• Incorrectly applying the power of a power rule: A common mistake is to misapply the power of a power rule, such as thinking (a^m)n = a^(m+n) instead of (a^m)n = a^(mn). Always double-check the rule before applying it.
• Forgetting to distribute when multiplying exponents: When multiplying an exponent by an expression, remember to distribute the multiplication to all terms within the expression. For example, in the equation (3^(-1))(3x+6), you need to multiply -1 by both 3x and 6.
• Equating exponents before having the same base: You can only equate exponents if the bases are the same. Make sure to express both sides of the equation with the same base before equating the exponents.
• Making arithmetic errors when solving the linear equation: After equating the exponents, you'll typically end up with a linear equation. Be careful when solving this equation and avoid simple arithmetic errors.
• Not verifying the solution: As mentioned earlier, verifying the solution is crucial to ensure its correctness. Always substitute the solution back into the original equation to check if it holds true.

By being mindful of these common mistakes, you can significantly reduce the chances of errors and improve your ability to solve exponential equations accurately. Remember, practice and attention to detail are key to mastering these types of problems. Let's move on to some additional practice problems to further solidify your understanding.

Additional Practice Problems

To solidify your understanding of solving exponential equations, let's tackle a few more practice problems. These problems will provide you with the opportunity to apply the techniques we've discussed and further develop your problem-solving skills. Remember, the key to mastering any mathematical concept is practice, practice, practice!

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

These problems cover a range of scenarios, including different bases and exponents. Work through each problem step-by-step, applying the techniques we've discussed, such as expressing both sides with the same base, applying the power of a power rule, equating exponents, and solving the resulting linear equation. Don't forget to verify your solutions to ensure accuracy. As you work through these problems, pay attention to the nuances of each equation and the specific steps required to solve it. This will help you develop a deeper understanding of exponential equations and improve your problem-solving abilities. If you encounter any difficulties, revisit the step-by-step solution we provided earlier and review the key concepts and properties of exponents. Remember, the goal is not just to find the correct answer but also to understand the underlying principles and develop a systematic approach to solving these types of problems. With consistent practice, you'll become more confident and proficient in solving exponential equations.

Conclusion

In this comprehensive guide, we've explored the process of solving the exponential equation (1/3)^(3x+6) = 9^(x-2). We've delved into the fundamental concepts of exponential equations, discussed the key properties of exponents, and provided a detailed step-by-step solution. We've also highlighted common mistakes to avoid and offered additional practice problems to solidify your understanding. Solving exponential equations is a fundamental skill in mathematics with applications in various fields. By mastering the techniques discussed in this guide, you'll be well-equipped to tackle a wide range of exponential problems. Remember, the key to success in mathematics is practice and a thorough understanding of the underlying concepts. Don't be afraid to challenge yourself with more complex problems and to seek help when needed. With consistent effort, you can develop the skills and confidence to excel in solving exponential equations and other mathematical challenges. The journey of learning mathematics is a continuous one, and each problem solved is a step forward in your mathematical journey. So, keep practicing, keep exploring, and keep challenging yourself, and you'll be amazed at what you can achieve. Congratulations on taking the time to learn and improve your skills in solving exponential equations! We hope this guide has been helpful and informative. Now, go forth and conquer those exponential equations!