Solve The Exponential Equation 8^(4x+1) = 64

Exponential equations can often seem daunting at first glance, but with a systematic approach and a solid understanding of exponent rules, they become quite manageable. This article will delve into the process of solving the equation 8^(4x+1) = 64, providing a comprehensive, step-by-step explanation to ensure clarity and understanding. Our focus will be on transforming the equation into a solvable form by utilizing the properties of exponents and logarithms. We aim to not only find the solution but also to equip you with the knowledge to tackle similar problems with confidence.

Understanding the Fundamentals of Exponential Equations

Before diving into the specific problem, let's establish a foundation in exponential equations. At their core, exponential equations involve variables in the exponents. The key to solving these equations lies in manipulating them to a form where we can directly compare the exponents. This is often achieved by expressing both sides of the equation with the same base. When the bases are the same, the exponents must be equal for the equation to hold true. For instance, if we have a^m = a^n, then it directly follows that m = n. This principle forms the backbone of our strategy. Additionally, understanding the properties of exponents, such as the power of a power rule ((a^m)n = a^(m*n)) and the product of powers rule (a^m * a^n = a^(m+n)), is crucial. These rules allow us to simplify and manipulate exponential expressions, bringing us closer to the solution. In the context of our problem, 8^(4x+1) = 64, we need to recognize that both 8 and 64 can be expressed as powers of the same base, which is 2. This realization is the first step in our solution process. Recognizing these underlying principles and having them readily available in our toolkit is what makes solving exponential equations not just a matter of applying formulas, but a process of logical deduction and strategic manipulation. We will consistently refer back to these fundamentals as we progress through the solution, ensuring that each step is grounded in sound mathematical principles.

Step 1: Expressing Both Sides with a Common Base

In tackling the equation 8^(4x+1) = 64, the crucial first step is to identify a common base for both sides. Recognizing that both 8 and 64 are powers of 2 is key. We can express 8 as 2^3 and 64 as 2^6. Rewriting the equation with this common base transforms it from 8^(4x+1) = 64 to (2³⁾(4x+1) = 2^6. This transformation is significant because it allows us to utilize the properties of exponents more effectively. Specifically, the power of a power rule, which states that (a^m)n = a^(mn), comes into play. Applying this rule to the left side of the equation, we multiply the exponents 3 and (4x+1), resulting in 2^(3(4x+1)) = 2^6. This simplification is a pivotal moment in the solution process. By expressing both sides of the equation with the same base, we've set the stage for a direct comparison of the exponents. This step not only simplifies the equation but also makes it more intuitive to solve. The complexity of dealing with different bases is removed, and the problem is now focused on the exponents themselves. The ability to recognize and utilize common bases is a fundamental skill in solving exponential equations, and mastering this technique opens the door to solving a wide range of similar problems. As we move forward, we'll see how this initial transformation streamlines the remaining steps, ultimately leading us to the solution for x.

Step 2: Equating the Exponents

With the equation now expressed as 2^(3(4x+1)) = 2^6*, the next logical step is to equate the exponents. This is a direct consequence of the fundamental principle that if a^m = a^n, then m = n. In our case, this principle allows us to set the exponents equal to each other, transforming the exponential equation into a simpler algebraic equation. Therefore, we can write 3*(4x+1) = 6. This equation is much easier to handle compared to the original exponential form. It's a linear equation in one variable, which we can solve using standard algebraic techniques. The act of equating the exponents is a crucial bridge between the world of exponential equations and the familiar territory of algebraic equations. It's a powerful technique that simplifies the problem considerably. By focusing on the exponents, we bypass the complexities of the exponential expressions and work with a more manageable equation. This step highlights the elegance of mathematical problem-solving, where a strategic transformation can significantly reduce the difficulty of a problem. As we proceed, we will see how solving this linear equation will directly lead us to the value of x, the solution to our original exponential equation. The ability to make this transition from exponential to algebraic form is a key skill in mathematics, applicable in various contexts beyond just solving equations.

Step 3: Solving the Linear Equation

Now that we have the linear equation 3(4x+1) = 6*, the task is to solve for x. This involves a series of algebraic manipulations to isolate x on one side of the equation. First, we distribute the 3 on the left side, which gives us 12x + 3 = 6. Next, we subtract 3 from both sides of the equation to isolate the term with x, resulting in 12x = 3. Finally, we divide both sides by 12 to solve for x, which yields x = 3/12. This fraction can be further simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 3. This simplification gives us x = 1/4. Each of these steps is a fundamental algebraic operation, and their careful application is essential for arriving at the correct solution. Solving linear equations is a cornerstone of algebra, and this step demonstrates the importance of mastering these basic skills. The process of isolating a variable involves undoing the operations that are applied to it, following the reverse order of operations. In this case, we undid the multiplication by 3, the addition of 3, and finally, the multiplication by 12. The ability to systematically apply these operations is crucial for success in algebra and beyond. As we have found the value of x, we are now one step away from completing the solution to the original exponential equation. The next step will involve verifying our solution to ensure its accuracy.

Step 4: Expressing the Solution in Simplest Form

Having found x = 3/12, it is essential to express this solution in its simplest form. Simplification is a crucial step in mathematics, ensuring that the answer is presented in the most concise and easily understandable manner. In the case of fractions, simplification involves reducing the fraction to its lowest terms. This is achieved by dividing both the numerator and the denominator by their greatest common divisor (GCD). For the fraction 3/12, the GCD of 3 and 12 is 3. Dividing both the numerator and the denominator by 3, we get (3 ÷ 3) / (12 ÷ 3) = 1/4. Therefore, the simplified form of the solution is x = 1/4. Expressing solutions in simplest form is not just about aesthetics; it also makes the solution easier to work with in further calculations or applications. A simplified fraction is less prone to errors in subsequent computations. Moreover, it demonstrates a thorough understanding of mathematical principles and a commitment to precision. The process of simplification often involves recognizing common factors and applying division to reduce the numbers involved. This skill is fundamental in various areas of mathematics, from basic arithmetic to advanced calculus. In the context of our problem, simplifying the fraction not only provides the final answer but also reinforces the importance of mathematical rigor and attention to detail. Now that we have the simplified solution, we can confidently state the value of x that satisfies the original equation.

Final Answer: x = 1/4

In conclusion, the solution to the equation 8^(4x+1) = 64 is x = 1/4. We arrived at this answer by systematically applying the principles of exponential equations and algebraic manipulation. The process began with expressing both sides of the equation using a common base, which allowed us to equate the exponents. This transformed the exponential equation into a linear equation, which we then solved for x. Finally, we simplified the solution to its simplest form. This step-by-step approach not only provides the correct answer but also demonstrates a clear and logical method for solving similar problems. The ability to solve exponential equations is a valuable skill in mathematics, with applications in various fields such as science, engineering, and finance. The key to success lies in understanding the properties of exponents, recognizing common bases, and applying algebraic techniques with precision. By mastering these skills, one can confidently tackle a wide range of exponential equations. The solution x = 1/4 represents the value that, when substituted back into the original equation, makes the equation true. This verification is an essential step in problem-solving, ensuring the accuracy of the solution. The journey from the initial equation to the final answer highlights the power of mathematical reasoning and the importance of a methodical approach. With practice and a solid understanding of the underlying principles, solving exponential equations can become a manageable and even enjoyable task.