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

Hey guys! Today, we're diving into a fun math problem where we need to figure out the value of 'a' in the equation (1/7)^(3a+3) = 343^(a-1). This might look a little intimidating at first, but don't worry, we'll break it down step by step and make it super easy to understand. Math can be like a puzzle, and we're here to put the pieces together!

Understanding the Problem

Before we jump into solving, let's make sure we really understand what the problem is asking. We have an equation with exponents, and our mission is to find the value of the variable 'a' that makes the equation true. Remember, equations are like a balance – what's on one side must equal what's on the other. Our job is to manipulate the equation until we isolate 'a' and find its value.

Key Concepts to Keep in Mind:

• Exponents: An exponent tells us how many times to multiply a number by itself. For example, 7^2 (7 squared) means 7 * 7 = 49.
• Negative Exponents: A negative exponent means we take the reciprocal of the base raised to the positive exponent. For example, 7^(-1) is the same as 1/7.
• Fractional Exponents: We won't directly deal with fractional exponents in this problem, but it's good to know they represent roots. For example, 7^(1/2) is the square root of 7.
• Laws of Exponents: These are rules that help us simplify expressions with exponents. We'll use some of these rules as we solve the problem.

Initial Assessment

Looking at the equation (1/7)^(3a+3) = 343^(a-1), we can see a few things right away:

1. We have fractions and whole numbers involved.
2. The bases (1/7 and 343) look related. In fact, 343 is a power of 7 (343 = 777 = 7^3).
3. We have expressions with 'a' in the exponents.

Our strategy will be to rewrite the equation so that both sides have the same base. This will allow us to equate the exponents and solve for 'a'. Let's get started!

Step-by-Step Solution

1. Express Both Sides with the Same Base

This is the crucial first step. We need to rewrite both sides of the equation using the same base. Since we know 343 is 7 cubed (7^3) and 1/7 is 7 to the power of -1 (7^-1), we can rewrite the equation as follows:

• (1/7)^(3a+3) = (7^(-1))(3a+3)
• 343^(a-1) = (7³⁾(a-1)

So our equation now looks like this:

(7^(-1))(3a+3) = (7³⁾(a-1)

2. Apply the Power of a Power Rule

Here's where our exponent rules come in handy! The power of a power rule states that (x^m)n = x^(m*n). This means when we raise a power to another power, we multiply the exponents. Let's apply this rule to both sides of our equation:

• (7^(-1))(3a+3) = 7^[1] = 7^(-3a-3)
• (7³⁾(a-1) = 7^[2] = 7^(3a-3)

Now our equation is significantly simpler:

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

3. Equate the Exponents

Since the bases are the same (both are 7), for the equation to hold true, the exponents must be equal. This is a fundamental concept in solving exponential equations. So, we can set the exponents equal to each other:

-3a - 3 = 3a - 3

4. Solve for 'a'

Now we have a simple linear equation to solve. Let's get all the 'a' terms on one side and the constants on the other. First, let's add 3a to both sides:

-3a - 3 + 3a = 3a - 3 + 3a

-3 = 6a - 3

Next, add 3 to both sides:

-3 + 3 = 6a - 3 + 3

0 = 6a

Finally, divide both sides by 6:

0 / 6 = 6a / 6

0 = a

So, we've found that a = 0. Hooray!

Verification

It's always a good idea to check our answer to make sure it's correct. Let's plug a = 0 back into the original equation:

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

(1/7)^(3(0)+3) = 343^(0-1)

(1/7)^3 = 343^(-1)

(1/7)^3 = 1/343

1/(7^3) = 1/343

1/343 = 1/343

Our solution checks out! Both sides of the equation are equal when a = 0.

Conclusion

So, guys, we've successfully solved for 'a' in the equation (1/7)^(3a+3) = 343^(a-1). We found that a = 0 is the value that makes the equation true. Remember, the key to solving exponential equations like this is to express both sides with the same base, then equate the exponents and solve the resulting equation. Keep practicing, and you'll become a math whiz in no time! Remember that understanding the laws of exponents is crucial here. Using exponent rules simplifies the process, and verifying the solution ensures accuracy. This step-by-step method can be applied to various similar problems.

This problem illustrates the importance of rewriting expressions to have a common base. Once you identify the relationship between the bases, the rest of the solution flows naturally. Remember, practice makes perfect, so tackle more problems like this to build your confidence and skills. In this case, understanding that 343 is a power of 7 is fundamental to simplifying the equation. The ability to recognize such relationships is a key skill in algebra. And always, remember to verify your solution! Plugging the value back into the original equation confirms that your answer is correct. This not only gives you confidence but also helps you catch any potential errors. Keep exploring math, and you'll uncover its many fascinating aspects! Understanding the underlying principles and practicing regularly will make you a confident problem solver. Remember, every math problem is a puzzle waiting to be solved! The sense of accomplishment you get from finding the solution is truly rewarding. So, keep challenging yourself, and enjoy the journey of learning mathematics.

I hope this explanation was clear and helpful. If you have any more questions or want to tackle other math problems, just let me know. Happy problem-solving!