Solving Exponential Equations A Step-by-Step Guide For $(\sqrt{7})^{6x} = 49^{x-6}$

Introduction

In this article, we will delve into the step-by-step solution of the exponential equation $(\sqrt{7})^{6x} = 49^{x-6}$. Exponential equations, which involve variables in the exponents, often require a strategic approach to solve. The key to tackling such equations lies in expressing both sides of the equation with a common base. This allows us to equate the exponents and solve for the unknown variable. This article aims to provide a comprehensive understanding of the methods used to solve this specific equation, while also offering valuable insights into the broader techniques applicable to solving exponential equations in general. Understanding exponential equations is crucial for various fields such as mathematics, physics, engineering, and finance, where exponential models are frequently used to describe growth and decay processes. We will explore how to manipulate the equation, apply exponent rules, and ultimately arrive at the solution. By the end of this discussion, you will have a solid understanding of how to approach and solve similar exponential equations effectively. In addition to the specific solution, we will also touch upon the underlying principles and properties of exponents that make this method possible. This includes the power of a power rule, the product of powers rule, and the quotient of powers rule. These rules are the foundational building blocks for simplifying and solving exponential expressions and equations. Moreover, we will discuss the importance of checking the solution obtained to ensure its validity, as extraneous solutions can sometimes arise when dealing with exponential equations. By carefully verifying the solution, we can confirm that it satisfies the original equation and is indeed a valid solution.

Rewriting the Equation with a Common Base

The first step in solving the equation $(\sqrt{7})^{6x} = 49^{x-6}$ is to express both sides using the same base. We observe that both $\sqrt{7}$ and $49$ can be written as powers of $7$. Specifically, $\sqrt{7}$ is equivalent to $7^{\frac{1}{2}}$, and $49$ is equivalent to $7^2$. By rewriting the equation in terms of the base $7$, we can simplify the equation and make it easier to solve. This is a common strategy when dealing with exponential equations: finding a common base allows us to directly compare the exponents. The process of rewriting involves applying the properties of exponents, which state that $(a^m)^n = a^{mn}$. Understanding and applying these properties is crucial for manipulating exponential expressions effectively. The ability to identify a common base is a fundamental skill in solving exponential equations, and it often involves recognizing the prime factorization of the numbers involved. In this case, recognizing that both $\sqrt{7}$ and $49$ are powers of $7$ is the key to simplifying the equation. Once we have rewritten the equation with a common base, we can then equate the exponents and solve the resulting linear equation. This transformation of the equation is a significant step towards finding the solution, as it converts the exponential equation into a more manageable algebraic form. Furthermore, the technique of expressing numbers with a common base extends beyond solving equations and is also used in simplifying exponential expressions and comparing their magnitudes. The strategic selection of the base can often lead to significant simplifications and a clearer understanding of the relationships between the numbers involved.

Let's rewrite the equation:

$(\sqrt{7})^{6x} = 49^{x-6}$ can be rewritten as $(7^{\frac{1}{2}})^{6x} = (7^2)^{x-6}$.

Applying the Power of a Power Rule

Next, we apply the power of a power rule, which states that $(a^m)^n = a^{mn}$. Applying this rule to both sides of the equation, we get $7^{\frac{1}{2} imes 6x} = 7^{2 imes (x-6)}$. Simplifying the exponents, we have $7^{3x} = 7^{2x-12}$. This step is crucial in solving exponential equations because it allows us to eliminate the bases and focus solely on the exponents. The power of a power rule is a fundamental property of exponents, and its application here demonstrates how it can simplify complex expressions. This rule essentially tells us that when raising a power to another power, we multiply the exponents. Understanding and being able to apply this rule is essential for manipulating exponential expressions and solving equations involving exponents. In this context, applying the rule transforms the equation from one with nested exponents to a simpler form where the exponents can be directly compared. The simplification achieved in this step is a direct result of the power of a power rule, which is one of the cornerstones of exponential algebra. Moreover, the application of this rule highlights the importance of understanding the order of operations in mathematics. It is crucial to correctly identify the exponents that need to be multiplied to simplify the expression accurately. This step-by-step application of exponent rules is a hallmark of solving exponential equations, and mastering these rules is key to success in this area of mathematics. Furthermore, this process not only simplifies the equation but also sets the stage for the next step, which involves equating the exponents.

Equating the Exponents

Since the bases are now the same, we can equate the exponents: $3x = 2x - 12$. This is a key step in solving exponential equations. When the bases are equal, the equation holds true only if the exponents are equal. This principle allows us to transform the exponential equation into a simpler algebraic equation, which in this case is a linear equation. The ability to equate exponents is a direct consequence of the properties of exponential functions. Specifically, if $a^m = a^n$, where $a$ is a positive number not equal to 1, then it must be the case that $m = n$. This property is fundamental to solving exponential equations and is widely used in various mathematical and scientific contexts. The process of equating exponents is a powerful tool because it allows us to eliminate the exponential terms and focus on solving a more manageable equation. In this particular case, equating the exponents results in a linear equation in one variable, which can be easily solved using standard algebraic techniques. The simplicity of this step underscores the importance of the initial steps of finding a common base and applying exponent rules, as these steps pave the way for this crucial simplification. Moreover, this step highlights the connection between exponential equations and algebraic equations, demonstrating how the former can be transformed into the latter through the application of mathematical principles. The transition from an exponential equation to a linear equation is a significant simplification and allows us to solve for the unknown variable using familiar algebraic methods.

Solving for x

Now, we solve the linear equation $3x = 2x - 12$. Subtracting $2x$ from both sides, we get $x = -12$. This is the solution to the exponential equation. The process of solving a linear equation involves isolating the variable on one side of the equation, which is achieved by performing the same operations on both sides to maintain the equality. In this case, subtracting $2x$ from both sides is a crucial step in isolating $x$. The simplicity of this solution step highlights the effectiveness of the previous steps in transforming the original exponential equation into a manageable form. Once the equation is reduced to a simple linear equation, the solution can be obtained with minimal effort. This process demonstrates the power of algebraic manipulation in solving mathematical problems. The solution $x = -12$ represents the value that, when substituted back into the original equation, will make the equation true. This is the essence of solving an equation: finding the value or values of the variable that satisfy the equation. Furthermore, the solution to this equation provides insight into the behavior of exponential functions and how they can be used to model various phenomena. The negative value of the solution is not uncommon in exponential equations and reflects the nature of exponential growth and decay processes. In addition to finding the solution, it is always a good practice to check the solution by substituting it back into the original equation to ensure its validity.

Checking the Solution

To verify our solution, we substitute $x = -12$ back into the original equation: $(\sqrt{7})^{6(-12)} = 49^{-12-6}$. This gives us $(\sqrt{7})^{-72} = 49^{-18}$. Rewriting both sides as powers of $7$, we get $(7^{\frac{1}{2}})^{-72} = (7^2)^{-18}$. Applying the power of a power rule, we have $7^{-36} = 7^{-36}$, which confirms that our solution is correct. Checking the solution is a crucial step in the process of solving exponential equations as it helps to ensure that the solution obtained is valid and does not introduce any extraneous roots. Extraneous roots are solutions that arise during the solving process but do not satisfy the original equation. These can occur when performing operations that are not reversible, such as squaring both sides of an equation or taking logarithms. In the case of exponential equations, substituting the solution back into the original equation helps to verify that the exponents are indeed equal when the bases are the same. This verification process also serves as a double-check of the algebraic manipulations performed during the solving process. If the solution does not satisfy the original equation, it indicates that there might have been an error in the steps taken to solve the equation. The process of checking the solution involves careful substitution and application of exponent rules to ensure that both sides of the equation are equal. In this particular case, the verification confirms that $x = -12$ is indeed the correct solution to the exponential equation. This step provides confidence in the solution and reinforces the understanding of the underlying principles of exponential equations. Furthermore, the act of checking the solution reinforces the importance of precision and attention to detail in mathematical problem-solving.

Conclusion

In conclusion, the solution to the equation $(\sqrt{7})^{6x} = 49^{x-6}$ is $x = -12$. We solved this equation by rewriting both sides with a common base, applying the power of a power rule, equating the exponents, and solving the resulting linear equation. Finally, we verified our solution by substituting it back into the original equation. This systematic approach is applicable to a wide range of exponential equations. Solving exponential equations is a fundamental skill in mathematics with applications in various fields. The key to solving these equations lies in understanding and applying the properties of exponents, such as the power of a power rule, the product of powers rule, and the quotient of powers rule. These rules allow us to manipulate exponential expressions and simplify equations. The process of finding a common base is a crucial step in solving exponential equations, as it allows us to equate the exponents and transform the equation into a more manageable form. Furthermore, the ability to check the solution is essential to ensure its validity and avoid extraneous roots. The systematic approach outlined in this article provides a framework for solving exponential equations effectively. By mastering these techniques, you can confidently tackle a wide variety of exponential equations and apply them to real-world problems. The solution to this particular equation demonstrates the power of algebraic manipulation and the importance of understanding the properties of exponents. Moreover, the process of solving this equation highlights the connection between exponential functions and linear equations, showcasing how mathematical concepts are interconnected. The ability to solve exponential equations is a valuable skill that can be applied in various contexts, from scientific research to financial analysis. Mastering these techniques will enhance your problem-solving abilities and deepen your understanding of mathematics.