Solving For X In Exponential Equations (3/5)^(-6) = (3/5)^(3x) * (3/5)^(2x-1)

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Introduction

In mathematics, solving for variables within exponential equations is a fundamental skill. This article dives deep into how to find the value of x in the given equation: (3/5)^(-6) = (3/5)^(3x) * (3/5)^(2x-1). Understanding the properties of exponents is crucial for tackling such problems. We will break down the equation step-by-step, ensuring a clear and comprehensive explanation suitable for students and math enthusiasts alike. Our approach focuses on simplifying the equation using exponent rules, combining like terms, and ultimately isolating x to determine its value. This detailed guide will not only provide the solution but also enhance your understanding of exponential equations and their manipulations.

Understanding the Problem

Before we jump into the solution, let's make sure we fully grasp the problem. We are given an exponential equation where the base is a fraction, and the exponents involve the variable x. Our primary goal is to isolate x and find its numerical value. To achieve this, we will utilize the properties of exponents, which allow us to simplify and combine terms effectively. Specifically, we will use the rule that states when multiplying exponential terms with the same base, we can add their exponents. This rule is pivotal in simplifying the right-hand side of the equation, where we have two terms with the same base (3/5) multiplied together. Recognizing the structure of the equation and identifying the applicable exponent rules are essential first steps in solving any exponential equation. This initial understanding sets the stage for a systematic approach to finding the value of x.

Step-by-Step Solution

1. Apply the Product of Powers Rule

The first step in solving the equation (3/5)^(-6) = (3/5)^(3x) * (3/5)^(2x-1) is to simplify the right-hand side. We apply the product of powers rule, which states that a^(m) * a^(n) = a^(m+n). This allows us to combine the two exponential terms on the right. By adding the exponents 3x and (2x - 1), we simplify the equation. This rule is a cornerstone of exponent manipulation, allowing us to consolidate multiple terms into a single term, making the equation easier to handle. Applying this rule correctly is crucial for the subsequent steps in solving for x. The result of this step significantly reduces the complexity of the equation, bringing us closer to isolating the variable.

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

2. Simplify the Exponent

Following the application of the product of powers rule, we now simplify the exponent on the right-hand side of the equation. This involves combining like terms within the exponent 3x + (2x - 1). By adding 3x and 2x, we get 5x. The simplified exponent becomes 5x - 1. This step is a straightforward algebraic simplification, but it's crucial for reducing the equation to its simplest form. Simplification of exponents allows us to clearly see the relationship between the exponents on both sides of the equation, which is key to solving for x. The simplified equation now looks more manageable, setting the stage for the next step where we equate the exponents.

(3/5)^(-6) = (3/5)^(5x - 1)

3. Equate the Exponents

Since the bases on both sides of the equation are the same (3/5), we can equate the exponents. This is a direct application of the property that if a^(m) = a^(n), then m = n. Therefore, we set the exponent on the left side, -6, equal to the exponent on the right side, 5x - 1. This step transforms the exponential equation into a simple linear equation, which is much easier to solve. Equating exponents is a powerful technique in solving exponential equations, allowing us to eliminate the exponential part and focus on the algebraic relationship between the exponents. This transition is a critical step in isolating and finding the value of x.

-6 = 5x - 1

4. Solve for x

Now we solve the linear equation -6 = 5x - 1 for x. First, we add 1 to both sides of the equation to isolate the term with x. This gives us -5 = 5x. Next, we divide both sides by 5 to solve for x. This yields x = -1. This step involves basic algebraic manipulations to isolate the variable, and it is the final step in determining the value of x. The solution x = -1 satisfies the original exponential equation, making it the correct answer. This process demonstrates the effectiveness of using exponent rules and algebraic techniques to solve exponential equations.

-6 + 1 = 5x

-5 = 5x

x = -1

Verification

To ensure our solution is correct, we substitute x = -1 back into the original equation: (3/5)^(-6) = (3/5)^(3x) * (3/5)^(2x-1). Substituting x = -1, we get (3/5)^(-6) = (3/5)^(3(-1)) * (3/5)^(2(-1)-1). This simplifies to (3/5)^(-6) = (3/5)^(-3) * (3/5)^(-3). Applying the product of powers rule on the right side, we have (3/5)^(-6) = (3/5)^(-3 + -3), which simplifies to (3/5)^(-6) = (3/5)^(-6). Since both sides of the equation are equal, our solution x = -1 is verified. This verification step is crucial in confirming the accuracy of our solution and ensuring a thorough understanding of the problem-solving process. It reinforces the concepts of exponent manipulation and equation solving.

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

(3/5)^(-6) = (3/5)^(-3) * (3/5)^(-3)

(3/5)^(-6) = (3/5)^(-3 + -3)

(3/5)^(-6) = (3/5)^(-6)

Conclusion

In conclusion, we have successfully found the value of x in the equation (3/5)^(-6) = (3/5)^(3x) * (3/5)^(2x-1). By applying the properties of exponents, simplifying the equation, and solving the resulting linear equation, we determined that x = -1. This process highlights the importance of understanding exponent rules and algebraic techniques in solving mathematical problems. The step-by-step solution provided a clear and logical path from the initial equation to the final answer. Furthermore, the verification step ensured the accuracy of our solution. Mastering these skills is essential for advancing in mathematics and tackling more complex problems. The ability to manipulate exponential equations and solve for variables is a valuable asset in various fields of study and practical applications.

Keywords

Exponential equations, solving for x, exponent rules, product of powers rule, equating exponents, algebraic simplification, verification, mathematics, equation solving, step-by-step solution