Solve $(\sqrt{7})^{6x} = 49^{x-6}$: A Step-by-Step Guide

Hey guys! Today, we're diving into an exciting math problem that involves solving an exponential equation. Exponential equations might seem intimidating at first, but with a clear, step-by-step approach, they can become quite manageable. Our specific problem is $(\sqrt{7})^{6x} = 49^{x-6}$. This equation combines radicals and exponents, making it a perfect example to illustrate key techniques in solving such problems. We'll break down each step, ensuring you understand the logic behind every move. By the end of this guide, you’ll not only be able to solve this particular equation but also gain confidence in tackling similar challenges. So, let's grab our math hats and get started! We'll explore how to manipulate exponents, use the properties of radicals, and ultimately find the value of 'x' that satisfies the equation. Remember, the key to mastering mathematics is practice and understanding the fundamental principles. This problem is a fantastic opportunity to sharpen those skills and boost your problem-solving abilities. Let's make math less of a mystery and more of an adventure!

Understanding the Fundamentals of Exponential Equations

Before we jump into the solution, it's crucial to understand the fundamentals of exponential equations. These equations involve variables in the exponents, which means our goal is to isolate the variable in the exponent. The basic form of an exponential equation is $a^{f(x)} = a^{g(x)}$, where 'a' is the base and $f(x)$ and $g(x)$ are expressions involving 'x'. The core principle we'll use here is that if we have the same base on both sides of the equation, we can equate the exponents. Think of it like this: if $2^x = 2^3$, then 'x' must be 3. This simple rule is the cornerstone of solving exponential equations. However, often the bases are not immediately the same, as in our problem $(\sqrt{7})^{6x} = 49^{x-6}$. This is where we need to manipulate the equation using exponent rules and properties of radicals. We need to express both sides of the equation with the same base. This often involves recognizing common bases and rewriting numbers as powers of those bases. For instance, in our case, we can express both $\sqrt{7}$ and 49 as powers of 7. This initial step is vital because it sets the stage for equating the exponents. Understanding these foundational concepts is key to not just solving this problem, but also to tackling a wide range of exponential equations. So, let's keep these principles in mind as we proceed with the step-by-step solution.

Step 1: Expressing Both Sides with the Same Base

The first critical step in solving $(\sqrt{7})^{6x} = 49^{x-6}$ is to express both sides of the equation using the same base. Looking at the equation, we see $\sqrt{7}$ on the left side and 49 on the right side. We need to find a common base for both. We know that 49 is a power of 7, specifically $49 = 7^2$. Also, we can express $\sqrt{7}$ as $7^{\frac{1}{2}}$. Now we have a common base of 7! Let's rewrite the equation using these expressions. The left side, $(\sqrt{7})^{6x}$, becomes $(7^{\frac{1}{2}})^{6x}$. Using the power of a power rule, which states that $(a^m)^n = a^{m*n}$, we can simplify this to $7^{\frac{1}{2} * 6x} = 7^{3x}$. On the right side, $49^{x-6}$ becomes $(7^2)^{x-6}$. Again, applying the power of a power rule, we get $7^{2*(x-6)} = 7^{2x-12}$. Now our equation looks much simpler: $7^{3x} = 7^{2x-12}$. This transformation is crucial because it allows us to move to the next step, which involves equating the exponents. By expressing both sides with the same base, we've essentially eliminated the exponential part and converted the problem into a simpler algebraic equation. Remember, identifying the common base is often the trickiest part, but once you've done that, the rest of the solution usually falls into place more easily. So, let's move on to the next step and see how we can use this simplified equation to solve for 'x'.

Step 2: Equating the Exponents

Now that we've successfully expressed both sides of the equation with the same base, 7, we have $7^{3x} = 7^{2x-12}$. This is where the fundamental principle of exponential equations comes into play: if $a^m = a^n$, then $m = n$. In other words, if two exponential expressions with the same base are equal, then their exponents must be equal. Applying this principle to our equation, we can equate the exponents: $3x = 2x - 12$. We've now transformed the exponential equation into a simple linear equation! This is a huge step forward because linear equations are much easier to solve. Our next goal is to isolate 'x' on one side of the equation. To do this, we can subtract 2x from both sides of the equation. This gives us $3x - 2x = 2x - 12 - 2x$, which simplifies to $x = -12$. And just like that, we've found the value of 'x'! Equating the exponents is a powerful technique that allows us to bypass the complexities of exponential expressions and work with simpler algebraic equations. It's a key step in solving many exponential equations, and mastering it will significantly enhance your problem-solving skills. Now that we have a potential solution, it's always a good idea to verify it to make sure it satisfies the original equation. Let's move on to the final step and check our answer.

Step 3: Solving the Linear Equation and Verifying the Solution

In the previous step, we equated the exponents and arrived at the linear equation $3x = 2x - 12$. By subtracting 2x from both sides, we found that $x = -12$. Now, let's verify this solution to ensure it satisfies the original equation, $(\sqrt{7})^{6x} = 49^{x-6}$. To verify, we substitute $x = -12$ back into the original equation. On the left side, we have $(\sqrt{7})^{6*(-12)} = (\sqrt{7})^{-72}$. Recall that $\sqrt{7} = 7^{\frac{1}{2}}$, so this becomes $(7^{\frac{1}{2}})^{-72} = 7^{\frac{1}{2} * -72} = 7^{-36}$. On the right side, we have $49^{x-6} = 49^{-12-6} = 49^{-18}$. Since $49 = 7^2$, this becomes $(7^2)^{-18} = 7^{2 * -18} = 7^{-36}$. Notice that both sides of the equation simplify to $7^{-36}$ when $x = -12$. This confirms that our solution is correct! Verification is a crucial step in solving any equation, as it helps us catch potential errors and ensures that our solution is valid. In this case, it gives us confidence that $x = -12$ is indeed the correct solution to the exponential equation. We've successfully navigated through the problem, using the properties of exponents and radicals, equating exponents, and solving a linear equation. This step-by-step process is a valuable tool for tackling similar exponential equations.

Conclusion: Mastering Exponential Equations

Alright, guys, we've successfully solved the exponential equation $(\sqrt{7})^{6x} = 49^{x-6}$! We started by understanding the fundamentals of exponential equations, then systematically broke down the problem into manageable steps. We first expressed both sides of the equation with the same base, which allowed us to equate the exponents. This transformation led us to a simple linear equation, which we solved to find $x = -12$. Finally, we verified our solution, confirming that it satisfies the original equation. This journey highlights the importance of understanding exponent rules, properties of radicals, and the fundamental principle of equating exponents when the bases are the same. Solving exponential equations might seem challenging at first, but with practice and a clear step-by-step approach, you can conquer them! Remember, the key is to identify common bases, manipulate exponents, and transform the equation into a more manageable form. Don't be afraid to break down complex problems into smaller, more digestible steps. By mastering these techniques, you'll not only be able to solve exponential equations but also gain a deeper understanding of mathematical concepts. Keep practicing, stay curious, and you'll continue to improve your problem-solving skills. Math is like a puzzle – every problem is a new challenge waiting to be solved. So, keep challenging yourself and enjoy the journey of learning!