Solving (5/7)^(x^2) = (49/25): A Step-by-Step Guide

Hey guys! Let's dive into solving this interesting exponential equation: (5/7)^(x2) = (49/25). It might look a bit intimidating at first, but don't worry, we'll break it down step-by-step. We'll explore the necessary steps to find the solutions for x and, importantly, how to verify if your answers are correct. So, grab your thinking caps, and let's get started!

Understanding Exponential Equations

Before we jump into the solution, it's crucial to understand what exponential equations are all about. In essence, exponential equations involve variables in the exponents. They are powerful tools in mathematics, often used to model growth and decay in various real-world scenarios, from population dynamics to radioactive decay. Understanding the underlying principles is key to tackling these problems effectively. When we talk about solving exponential equations, we're essentially trying to find the value(s) of the variable that make the equation true. This often involves manipulating the equation to get the variable by itself, which might require using logarithms, properties of exponents, or other algebraic techniques. Exponential equations are fundamental in many areas of science and engineering, so mastering them is a valuable skill.

The Basics of Exponents

To effectively tackle exponential equations, you've got to have a solid grasp of the basics of exponents. Remember, an exponent tells you how many times to multiply a base by itself. For example, in the expression 2^3, 2 is the base and 3 is the exponent, meaning you multiply 2 by itself three times (2 * 2 * 2 = 8). Understanding exponent rules is super important. One key rule is that when you have the same base raised to different powers and you multiply them, you add the exponents (e.g., x^a * x^b = x^(a+b)). Similarly, when you divide, you subtract the exponents (e.g., x^a / x^b = x^(a-b)). Another crucial rule for solving exponential equations is the power of a power rule: (x^a)b = x^(a*b). We'll be using this quite a bit! Lastly, remember that any number (except 0) raised to the power of 0 is 1 (x^0 = 1). These basic exponent rules are the building blocks for solving more complex exponential equations. So, before moving on, make sure you're comfortable with these concepts!

Step-by-Step Solution

Alright, let's get to the heart of the matter and solve the equation (5/7)^(x2) = (49/25). We're going to take a structured approach, making sure each step is clear and understandable. By following these steps, you'll not only solve this particular equation but also gain a solid methodology for tackling similar problems in the future.

Step 1: Express Both Sides with the Same Base

The first, and often most critical step in solving exponential equations, is to try and express both sides of the equation with the same base. This allows us to directly compare the exponents. Looking at our equation, (5/7)^(x2) = (49/25), we see that 49/25 can be rewritten as (7/5)^2. Now, we need to get creative to make the bases match. Notice that 7/5 is the reciprocal of 5/7. Remember, a reciprocal raised to a negative power gives us the original fraction. So, we can rewrite (7/5)^2 as (5/7)^(-2). This is a key manipulation! Our equation now looks like this: (5/7)^(x2) = (5/7)^(-2). By expressing both sides with the same base (5/7), we've set ourselves up to equate the exponents in the next step. This technique is fundamental in solving a wide range of exponential equations, so make sure you understand why and how we did this!

Step 2: Equate the Exponents

Now that we have the same base on both sides of the equation, (5/7)^(x2) = (5/7)^(-2), we can move on to the next logical step: equating the exponents. This is a powerful technique that works because if a^m = a^n, then m = n (provided a is not 0, 1, or -1). In our case, this means we can directly set the exponents equal to each other: x^2 = -2. This is a significant simplification! We've transformed the exponential equation into a much simpler quadratic equation. However, before we rush to solve it, let's pause for a moment and think about what this means. We've essentially removed the exponential part of the equation and are now dealing with a standard algebraic problem. This step highlights the core strategy in solving exponential equations: reducing them to simpler forms that we already know how to handle. This is where your knowledge of algebraic techniques comes into play, as we'll need to use them to find the values of x that satisfy this new equation.

Step 3: Solve the Quadratic Equation

We've arrived at a simple quadratic equation: x^2 = -2. Now, let's solve it! Remember, solving a quadratic equation means finding the values of x that make the equation true. In this case, we need to find the numbers that, when squared, equal -2. At first glance, you might notice something interesting. We're looking for a real number that, when multiplied by itself, results in a negative number. But wait a minute... the square of any real number is always non-negative! This means there are no real solutions to this equation. The solutions will be in the complex number system. To find them, we take the square root of both sides: x = ±√(-2). We can rewrite √(-2) as √(2 * -1) = √(2) * √(-1). Since √(-1) is defined as i (the imaginary unit), our solutions are x = ±√2 * i. So, we have two complex solutions: x = √2 * i and x = -√2 * i. This is a critical point! Not all exponential equations have real solutions, and this example beautifully illustrates that. When solving equations, always be mindful of the domain of the solutions you're looking for – in this case, we found complex solutions because there were no real ones.

Verifying the Solutions

Verifying your solutions is a crucial step in any mathematical problem, and it's especially important in exponential equations. It helps you catch any errors you might have made along the way and gives you confidence in your answers. Now that we've found our solutions (x = √2 * i and x = -√2 * i), let's plug them back into the original equation (5/7)^(x2) = (49/25) to see if they hold true. This process will not only confirm our solutions but also deepen our understanding of how these equations work.

The Importance of Verification

Before we dive into the verification process, let's take a moment to emphasize why it's so important. Verification acts as a safety net, catching errors that might have slipped through during the solving process. It's like a quality control check for your work. By plugging your solutions back into the original equation, you're essentially asking,