Solving Exponential Equations: Find X In $8^{x+2} - 8^{x+3} = 0$

$$

Hey guys! Let's dive into solving an interesting exponential equation today. We're going to tackle the equation $8^{x+2} - 8^{x+3} = 0$ and find out what the value of x is. If you've ever felt a bit puzzled by exponents, don't worry! We'll break it down step-by-step so it's super clear and easy to follow. Think of this as leveling up your math skills – are you ready? Let’s get started!

Understanding the Problem

Before we jump into solving, let's make sure we really get what the problem is asking. We have an equation where the unknown, x, is part of the exponent. This means we're dealing with an exponential equation. Exponential equations might seem a little intimidating at first, but they're actually quite manageable once you understand the basic rules of exponents.

Our main goal here is crystal clear: we need to isolate x and figure out what numerical value makes the equation true. To do this, we're going to use some cool properties of exponents and a little bit of algebraic magic. Remember, in math, we're like detectives, piecing together clues until we crack the case. This equation is our mathematical mystery, and we're about to solve it! So, keep your thinking caps on, and let’s move forward together.

Initial Equation

The equation we're tackling is:

$8^{x+2} - 8^{x+3} = 0$

This might look a bit tricky at first, but trust me, it's totally solvable. Our first step is to make it look a little simpler. We're going to use some exponent rules to rewrite the terms. Remember, the key to solving these equations is often to manipulate them into a form we can easily work with. So, let’s get started with the first step in simplifying this equation!

Step-by-Step Solution

Okay, let's get our hands dirty and solve this equation step by step. Remember, the key to any math problem is breaking it down into smaller, more manageable chunks. We're going to do just that with our equation $8^{x+2} - 8^{x+3} = 0$.

Step 1: Isolate the Exponential Terms

Our initial goal is to get the exponential terms on opposite sides of the equation. This will help us see the relationship between them more clearly. We can do this by adding $8^{x+3}$ to both sides of the equation. Think of it like balancing a scale – whatever we do to one side, we have to do to the other to keep things equal.

So, here’s what it looks like:

$8^{x+2} - 8^{x+3} + 8^{x+3} = 0 + 8^{x+3}$

This simplifies to:

$8^{x+2} = 8^{x+3}$

Now we have a much cleaner equation to work with. Both sides have an exponential term, which is perfect for our next step. We're on our way to cracking this problem – keep going!

Step 2: Use Properties of Exponents to Simplify

Now comes the fun part where we get to use some cool exponent rules! Remember, one of the key properties is that $a^{m+n} = a^m imes a^n$. We can use this to break down the terms in our equation and make them easier to compare.

Let's focus on the right side of the equation, $8^{x+3}$. We can rewrite this using our exponent rule:

$8^{x+3} = 8^{x} imes 8^{3}$

Similarly, we can rewrite the left side, $8^{x+2}$:

$8^{x+2} = 8^{x} imes 8^{2}$

Now, let's substitute these back into our equation. We started with $8^{x+2} = 8^{x+3}$, and now we have:

$8^{x} imes 8^{2} = 8^{x} imes 8^{3}$

See how we've broken down the exponents into simpler terms? This is going to make the next step much easier. Stick with me, we're getting closer to the solution!

Step 3: Divide Both Sides by a Common Factor

Okay, we're at a really interesting point in our solution. We've got the equation $8^{x} imes 8^{2} = 8^{x} imes 8^{3}$. Notice anything similar on both sides? That's right, we have $8^{x}$ as a common factor. This is super helpful because we can divide both sides by $8^{x}$ to simplify the equation even further.

But, before we do that, let's just take a quick detour to make sure we're not dividing by zero. We need to ask ourselves: can $8^{x}$ ever be zero? The answer is no! No matter what value we plug in for x, $8^{x}$ will never be zero. This is a crucial point because it means we're safe to divide without worrying about any mathematical mishaps.

So, let's go ahead and divide both sides by $8^{x}$:

rac{8^{x} imes 8^{2}}{8^{x}} = rac{8^{x} imes 8^{3}}{8^{x}}

The $8^{x}$ terms cancel out beautifully, leaving us with:

$8^{2} = 8^{3}$

Wait a second... This doesn't quite look right, does it? We know that $8^2$ (which is 64) is definitely not equal to $8^3$ (which is 512). This tells us that something interesting is happening, and we need to dig a little deeper. Hold tight, we're about to uncover the final piece of the puzzle!

Step 4: Revisit and Correct the Approach

Alright, detectives, let’s backtrack a tiny bit. We arrived at $8^{2} = 8^{3}$, which we know isn't true. This usually means we've hit a snag in our method, and it's time to re-evaluate our steps. It’s totally normal in math – sometimes the path isn't straight, and that’s okay! What's important is that we learn from it.

Let’s rewind to our equation $8^{x+2} = 8^{x+3}$. Instead of dividing by $8^x$, let’s try a different approach. The key here is to realize that for these two exponential terms to be equal, there has to be a specific condition. Think about it – when can two powers with the same base be equal?

One way to think about this is to bring everything to one side of the equation again. So, let’s go back to our original rearranged equation:

$8^{x+2} = 8^{x+3}$

Now, subtract $8^{x+3}$ from both sides to get:

$8^{x+2} - 8^{x+3} = 0$

We're back where we started in some ways, but this time, let’s factor out a common term. Factoring is like the reverse of distributing, and it’s a powerful tool in algebra. Can you spot a common factor in $8^{x+2}$ and $8^{x+3}$? Think about the smaller exponent.

We can factor out $8^{x+2}$. Let's see how that looks!

Step 5: Factor Out the Common Term

Okay, let’s roll up our sleeves and factor out the common term in our equation $8^{x+2} - 8^{x+3} = 0$. Remember, factoring helps us simplify complex expressions by pulling out the parts that are shared between terms. In this case, we identified that $8^{x+2}$ is a common factor.

So, let's factor $8^{x+2}$ out of both terms. This means we're dividing each term by $8^{x+2}$ and writing it outside a set of parentheses. Here’s how it works:

$8^{x+2}(1 - 8^{1}) = 0$

Notice how we factored out $8^{x+2}$ from both terms? When we divide $8^{x+2}$ by itself, we get 1. And when we divide $8^{x+3}$ by $8^{x+2}$, we're left with $8^{1}$ (since we subtract the exponents: $(x+3) - (x+2) = 1$).

Now, let's simplify the term inside the parentheses:

$8^{x+2}(1 - 8) = 0$

Which gives us:

$8^{x+2}(-7) = 0$

We're getting closer and closer to the finish line. This factored form is going to make it much easier to find the value of x. Let's see what our next move should be!

Step 6: Solve for x

We've arrived at the equation $8^{x+2}(-7) = 0$. Now, we need to figure out what value of x will make this equation true. Think about it: we have two factors multiplied together, and the result is zero. What does that tell us?

The key here is the zero product property. It says that if the product of two factors is zero, then at least one of the factors must be zero. In our case, we have two factors: $8^{x+2}$ and $(-7)$.

So, we can write:

$8^{x+2} = 0$ or $-7 = 0$

Well, $-7$ clearly does not equal $0$, so we can disregard that. This leaves us with:

$8^{x+2} = 0$

Now, this is a crucial point. We need to ask ourselves: is there any value of x that will make $8^{x+2}$ equal to zero? Remember, 8 raised to any power will never be zero. It will get incredibly close to zero as x becomes a large negative number, but it will never actually reach zero.

This means that there is no solution for x that satisfies the equation $8^{x+2} = 0$. So, after all our hard work, we've discovered something really important: this equation has no solution!

Conclusion

Wow, guys! We've taken quite the journey solving the exponential equation $8^{x+2} - 8^{x+3} = 0$. We broke it down step-by-step, used some cool exponent rules, factored out common terms, and even navigated a little detour when things didn't quite add up at first. That's the awesome thing about math – it's like a puzzle, and sometimes you have to try different approaches to find the solution.

In the end, we discovered that this equation has no solution. It's a fantastic reminder that not every equation has a neat and tidy answer, and that's perfectly okay! What’s important is that we learned a lot about exponential equations and problem-solving along the way. So, give yourselves a pat on the back for sticking with it – you've leveled up your math skills today!

Keep practicing, keep exploring, and remember, math is an adventure! You never know what cool discoveries you'll make next. Until then, happy solving!