Solving For A: (1/7)^(3a+3) = 343^(a-1)

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Solving for a: (1/7)^(3a+3) = 343^(a-1)

Hey guys! Let's dive into this interesting math problem where we need to find the value of 'a' that satisfies the equation (1/7)^(3a+3) = 343^(a-1). This might look a bit intimidating at first, but don't worry, we'll break it down step by step. This is a classic example of an exponential equation, and the key to solving these types of problems is to get both sides of the equation to have the same base. Once we have the same base, we can simply equate the exponents and solve for our variable. So, grab your thinking caps, and let's get started!

Understanding Exponential Equations

Before we jump into the solution, let's quickly recap what exponential equations are all about. An exponential equation is an equation in which the variable appears in the exponent. The general form of an exponential equation is something like a^x = b, where 'a' is the base, 'x' is the exponent, and 'b' is the result. Solving these equations often involves using logarithms or, as in this case, manipulating the bases to be the same. The fundamental principle we'll use here is that if a^m = a^n, then m = n. This allows us to transform a complex exponential equation into a simpler algebraic equation.

Remember, the goal is always to simplify the equation until we can isolate the variable we're trying to solve for. In our case, that's 'a'. To do this effectively, we'll need to use our knowledge of exponents and their properties. Understanding exponent rules such as the power of a power rule, the product of powers rule, and the quotient of powers rule is crucial for tackling these kinds of problems. Keep these rules in mind as we move forward, and you'll see how they help us make this problem much easier to handle.

Step-by-Step Solution

Step 1: Express Both Sides with the Same Base

The first crucial step is to express both sides of the equation with the same base. Notice that 343 is 7 cubed (7^3), and 1/7 is 7 to the power of -1 (7^-1). Rewriting the equation using the base 7 will help us simplify things significantly. So, let's rewrite our equation (1/7)^(3a+3) = 343^(a-1) using the base 7:

(7-1)(3a+3) = (73)(a-1)

This transformation is super important because now we can directly compare the exponents once we simplify further. This is a common technique when dealing with exponential equations, and mastering it will make solving these problems much smoother. Remember to always look for ways to express the numbers in terms of a common base, as it’s often the key to unlocking the solution.

Step 2: Simplify the Exponents

Now that we have the same base on both sides, we can simplify the exponents using the power of a power rule, which states that (am)n = a^(m*n). Applying this rule to our equation, we get:

7^(-1 * (3a+3)) = 7^(3 * (a-1))

Simplifying the exponents further, we have:

7^(-3a - 3) = 7^(3a - 3)

This step is essential because it transforms our equation into a form where we can directly compare the exponents. By applying the power of a power rule, we've eliminated the parentheses and made the exponents more manageable. Simplifying exponents is a fundamental skill in algebra, and it’s crucial for solving a wide range of mathematical problems, not just exponential equations.

Step 3: Equate the Exponents

Since the bases are the same, we can now equate the exponents. This is the core idea behind solving exponential equations when you can get the bases to match. If a^m = a^n, then m = n. So, in our case:

-3a - 3 = 3a - 3

This step is where the magic happens! We've turned a complex exponential equation into a simple linear equation. Equating exponents allows us to get rid of the exponential part and focus on solving for the variable directly. This is a powerful technique, and you'll use it frequently when dealing with exponential functions and equations.

Step 4: Solve for 'a'

Now we have a simple linear equation to solve for 'a'. Let's add 3a to both sides:

-3 = 6a - 3

Next, add 3 to both sides:

0 = 6a

Finally, divide both sides by 6:

a = 0

So, we've found our solution! The value of 'a' that satisfies the equation is 0. This step-by-step process demonstrates how to solve for 'a' in a clear and logical way. By isolating 'a', we've answered the question and completed the problem.

Final Answer

Therefore, the value of a that satisfies the equation (1/7)^(3a+3) = 343^(a-1) is 0. Wasn't that fun? By breaking down the problem into smaller, manageable steps, we were able to solve it quite easily. Remember the key strategies: get the same base, simplify the exponents, equate the exponents, and then solve the resulting equation. These skills will come in handy in many other math problems you'll encounter.

Practice Problems

To solidify your understanding, here are a few practice problems you can try:

  1. Solve for x: 2^(2x+1) = 8^(x-1)
  2. Find the value of y: (1/3)^(y+2) = 27^(y-1)
  3. What is z if: 5^(3z-2) = 125^(z+1)

Working through these problems will help you become more comfortable with solving exponential equations. Practice makes perfect, so don't hesitate to try them out and see how well you can apply the techniques we discussed.

Conclusion

In conclusion, solving exponential equations often involves manipulating the equation to have the same base on both sides, simplifying exponents, equating the exponents, and then solving the resulting algebraic equation. By following these steps, we successfully found that a = 0 for the given equation. Mastering these techniques will not only help you with exponential equations but also improve your overall problem-solving skills in mathematics. Keep practicing, and you'll become a math whiz in no time!