Solving Exponential Equations A Step By Step Guide For $243^{-y}=(\frac{1}{243})^{3 Y} \cdot 9^{-2 Y}$

Introduction

In this article, we delve into the process of solving an exponential equation, specifically focusing on the equation $243^{-y}=\left(\frac{1}{243}\right)^{3 y} \cdot 9^{-2 y}$. Exponential equations are a fundamental concept in algebra, and mastering the techniques to solve them is crucial for success in more advanced mathematical topics. Our goal is to find the value(s) of y that satisfy this equation. We will explore different methods, including simplifying expressions using exponent rules and converting all terms to a common base. By understanding these techniques, you'll be well-equipped to tackle a wide range of exponential equations. This article provides a detailed, step-by-step solution, ensuring clarity and understanding at each stage. We will also discuss the underlying principles and properties of exponents that make this solution possible. Let’s begin by examining the given equation and strategizing an approach to solve it efficiently. The equation involves terms with bases 243 and 9, which are both powers of 3, giving us a good starting point for simplification. Understanding the nature of exponential functions and their behavior is key to correctly manipulating and solving such equations. Furthermore, recognizing common bases and expressing all terms with the same base is a cornerstone technique in solving exponential equations. In the subsequent sections, we will elaborate on these aspects and demonstrate the solution process in detail. Remember, the ability to solve exponential equations is not just about finding the correct answer, it’s also about developing a strong foundation in algebraic manipulations and problem-solving strategies. This article aims to enhance both your understanding and your skills in this area.

Rewriting the Equation with a Common Base

The first step in solving the equation $243^{-y}=\left(\frac{1}{243}\right)^{3 y} \cdot 9^{-2 y}$ is to rewrite all terms using a common base. Recognizing that 243 and 9 are powers of 3 is crucial here. We can express 243 as $3^5$ and 9 as $3^2$. This allows us to rewrite the equation entirely in terms of base 3. This transformation is a common strategy when dealing with exponential equations, as it simplifies the terms and allows for easier manipulation. By expressing all terms with the same base, we can then equate the exponents, leading to a simpler algebraic equation. The original equation now becomes $(3^5)^{-y} = \left(\frac{1}{3^5}\right)^{3y} \cdot (3^2)^{-2y}$. This is a significant step because it converts the equation into a form where the exponents can be directly compared. We can further simplify the equation by rewriting $\frac{1}{3^5}$ as $3^{-5}$. This substitution is based on the property of exponents that states $a^{-n} = \frac{1}{a^n}$. With this change, our equation transforms into $(3^5)^{-y} = (3^{-5})^{3y} \cdot (3^2)^{-2y}$. Now, we can apply the power of a power rule, which states $(a^m)^n = a^{mn}$, to simplify the exponents. This rule is fundamental in simplifying exponential expressions and is used extensively in solving exponential equations. Applying this rule to each term, we get $3^{-5y} = 3^{-15y} \cdot 3^{-4y}$. The next step involves combining the terms on the right side of the equation, which will lead us to an even simpler form that we can solve for y. This methodical approach, breaking down the problem into smaller, manageable steps, is key to successfully solving exponential equations.

Simplifying the Equation Using Exponent Rules

Continuing from the previous step, we have the equation $3^{-5y} = 3^{-15y} \cdot 3^{-4y}$. To further simplify this, we need to use the product of powers rule, which states that $a^m \cdot a^n = a^{m+n}$. This rule allows us to combine the terms on the right side of the equation since they have the same base. Applying the product of powers rule, we get $3^{-5y} = 3^{-15y + (-4y)}$, which simplifies to $3^{-5y} = 3^{-19y}$. Now that we have both sides of the equation expressed with the same base, we can equate the exponents. This is a critical step in solving exponential equations, as it transforms the exponential equation into a simple algebraic equation. Equating the exponents gives us $-5y = -19y$. This is a linear equation in y, which is much easier to solve than the original exponential equation. To solve for y, we need to isolate y on one side of the equation. We can do this by adding 19y to both sides of the equation. This gives us $-5y + 19y = -19y + 19y$, which simplifies to $14y = 0$. Now, to find the value of y, we divide both sides of the equation by 14. This gives us $\frac{14y}{14} = \frac{0}{14}$, which simplifies to $y = 0$. Thus, we have found a solution for y. However, it's always a good practice to check our solution in the original equation to ensure it is valid. In the next section, we will verify this solution and discuss any potential extraneous solutions. The simplification process involving exponent rules is a cornerstone technique in solving not only exponential equations but also various algebraic problems.

Verifying the Solution

Now that we have found a potential solution, $y = 0$, it is crucial to verify this solution in the original equation, $243^{-y}=\left(\frac{1}{243}\right)^{3 y} \cdot 9^{-2 y}$. Verification is an essential step in solving any equation, as it helps us to ensure that our solution is correct and does not lead to any inconsistencies or undefined terms. Substituting $y = 0$ into the original equation, we get $243^{-0}=\left(\frac{1}{243}\right)^{3 (0)} \cdot 9^{-2 (0)}$. Any non-zero number raised to the power of 0 is 1, so $243^{-0} = 1$. Similarly, $\left(\frac{1}{243}\right)^{3 (0)} = \left(\frac{1}{243}\right)^{0} = 1$, and $9^{-2 (0)} = 9^{0} = 1$. Therefore, the equation becomes $1 = 1 \cdot 1$, which simplifies to $1 = 1$. This confirms that $y = 0$ is indeed a valid solution to the original equation. In this case, our verification process was straightforward. However, in some exponential equations, particularly those involving radicals or fractions in the exponents, verifying the solution can reveal extraneous solutions. Extraneous solutions are solutions that arise during the solving process but do not satisfy the original equation. This can happen when certain operations, such as squaring both sides of an equation, are performed. Therefore, the practice of verifying solutions is not just a formality, but a necessary step in ensuring the accuracy of the final result. In this instance, our solution $y = 0$ satisfies the original equation, making it the correct answer. Now, let's summarize our findings and conclude the solution process.

Conclusion

In summary, we have successfully solved the exponential equation $243^{-y}=\left(\frac{1}{243}\right)^{3 y} \cdot 9^{-2 y}$. The solution to this equation is $y = 0$. We arrived at this solution by following a systematic approach that involved rewriting the equation with a common base, simplifying using exponent rules, and verifying the solution. The key steps in our solution were: 1. Recognizing that 243 and 9 are powers of 3 and rewriting the equation in terms of base 3. 2. Applying the power of a power rule, $(a^m)^n = a^{mn}$, and the product of powers rule, $a^m \cdot a^n = a^{m+n}$, to simplify the equation. 3. Equating the exponents once both sides of the equation had the same base. 4. Solving the resulting linear equation for y. 5. Verifying the solution in the original equation to ensure its validity. This process highlights the importance of understanding and applying exponent rules in solving exponential equations. It also demonstrates the significance of verifying solutions to avoid extraneous results. Exponential equations are a common topic in algebra and calculus, and the techniques used here can be applied to a wide range of problems. By mastering these techniques, you can build a strong foundation in mathematics and improve your problem-solving skills. This particular problem illustrates the typical steps involved in solving exponential equations and reinforces the fundamental principles of exponents. We hope this detailed explanation has provided clarity and enhanced your understanding of solving exponential equations.

Answer

y=0