Solving $64^{-3x-3} ullet 64 + 22 = 38$ A Step-by-Step Guide

Introduction

Hey guys! Let's dive into solving the exponential equation 64^{-3x-3} ullet 64 + 22 = 38. Exponential equations might seem tricky at first, but with the right approach and understanding of the rules of exponents, they can be conquered! In this article, we'll break down the step-by-step solution to this equation, ensuring you grasp each concept along the way. We'll also sprinkle in some friendly tips and tricks to help you tackle similar problems in the future. So, buckle up and let's get started!

Understanding Exponential Equations

Before we jump into the nitty-gritty, let's quickly recap what exponential equations are all about. An exponential equation is simply an equation where the variable appears in the exponent. For example, in our equation 64^{-3x-3} ullet 64 + 22 = 38, the variable x is part of the exponent $-3x-3$. The key to solving these equations lies in manipulating them to isolate the exponential term and then using logarithms or properties of exponents to solve for the variable. Exponential equations pop up in various real-world scenarios, such as modeling population growth, radioactive decay, and compound interest. So, understanding how to solve them isn't just an academic exercise – it's a practical skill!

When dealing with exponential equations, it's crucial to remember some fundamental rules of exponents. These rules act as your toolbox, allowing you to simplify and rearrange the equation. Here are a few key ones:

1. Product of Powers: a^m ullet a^n = a^{m+n} (When you multiply powers with the same base, you add the exponents.)
2. Power of a Power: $(a^m)^n = a^{mn}$ (When you raise a power to another power, you multiply the exponents.)
3. Quotient of Powers: $a^m / a^n = a^{m-n}$ (When you divide powers with the same base, you subtract the exponents.)
4. Negative Exponent: $a^{-n} = 1/a^n$ (A negative exponent means you take the reciprocal of the base raised to the positive exponent.)
5. Zero Exponent: $a^0 = 1$ (Any non-zero number raised to the power of zero is 1.)

With these rules in mind, we're well-equipped to tackle our equation. Let's move on to the step-by-step solution.

Step-by-Step Solution to 64^{-3x-3} ullet 64 + 22 = 38

Step 1: Isolate the Exponential Term

The first thing we want to do is isolate the term with the exponent. In our equation, 64^{-3x-3} ullet 64 + 22 = 38, the exponential term is 64^{-3x-3} ullet 64. To isolate this term, we need to get rid of the $+22$ on the left side. We can do this by subtracting 22 from both sides of the equation:

64^{-3x-3} ullet 64 + 22 - 22 = 38 - 22

This simplifies to:

64^{-3x-3} ullet 64 = 16

Great! We've taken the first step towards solving our equation by isolating the exponential term. This is a crucial step in simplifying the equation and making it easier to work with. Remember, the goal here is to get the term with the variable exponent all by itself on one side of the equation. By isolating this term, we set ourselves up to use the properties of exponents and logarithms more effectively in the subsequent steps.

Step 2: Simplify Using Exponent Rules

Now that we've isolated the exponential term, let's simplify it using the rules of exponents. We have 64^{-3x-3} ullet 64 = 16. Notice that we have a product of powers with the same base (64). Remember the rule for the product of powers: a^m ullet a^n = a^{m+n}. We can apply this rule to our equation. First, let's rewrite 64 as $64^1$ for clarity:

64^{-3x-3} ullet 64^1 = 16

Now, apply the product of powers rule:

$64^{(-3x-3) + 1} = 16$

Simplify the exponent:

$64^{-3x-2} = 16$

This is a significant simplification! By applying the product of powers rule, we've reduced the complexity of the exponential term. The exponent is now a simpler expression, which makes the equation much easier to handle. This step showcases the power of understanding and applying exponent rules. It's not just about memorizing the rules, but also recognizing when and how to use them to make your life easier.

Step 3: Express Both Sides with the Same Base

To solve the equation further, we need to express both sides with the same base. We have $64^{-3x-2} = 16$. Notice that both 64 and 16 are powers of 4. Let's express both numbers in terms of base 4. We know that $64 = 4^3$ and $16 = 4^2$. Substituting these into our equation, we get:

$(4^3)^{-3x-2} = 4^2$

Now, we can use the power of a power rule, which states that $(a^m)^n = a^{mn}$. Applying this rule to the left side of the equation, we have:

$4^{3(-3x-2)} = 4^2$

Simplify the exponent on the left side:

$4^{-9x-6} = 4^2$

Why is this step so important? By expressing both sides of the equation with the same base, we've set the stage for a crucial simplification. When the bases are the same, we can equate the exponents, which transforms our exponential equation into a much simpler algebraic equation. This is a common and powerful technique in solving exponential equations. It's all about finding the common ground – the common base – that allows us to compare the exponents directly.

Step 4: Equate the Exponents

Now that we have the same base on both sides, we can equate the exponents. We have $4^{-9x-6} = 4^2$. Since the bases are equal, the exponents must be equal as well. So, we can write:

$-9x - 6 = 2$

This is a significant breakthrough! We've transformed our exponential equation into a simple linear equation. No more exponents to worry about – we're now in familiar algebraic territory. This step highlights the elegance of the method we're using. By strategically manipulating the equation and using the properties of exponents, we've managed to reduce a seemingly complex problem into a straightforward one. Equating exponents is a key technique that you'll find yourself using often when solving exponential equations.

Step 5: Solve for x

We now have a linear equation: $-9x - 6 = 2$. Let's solve for x. First, add 6 to both sides of the equation:

$-9x - 6 + 6 = 2 + 6$

This simplifies to:

$-9x = 8$

Now, divide both sides by -9:

$x = 8 / -9$

So, we have:

$x = -8/9$

And there you have it! We've successfully solved for x. This final step is the culmination of all our previous efforts. By carefully isolating the exponential term, applying exponent rules, expressing both sides with the same base, equating exponents, and then solving the resulting linear equation, we've navigated our way to the solution. This process demonstrates the power of a systematic approach to problem-solving. Each step builds upon the previous one, leading us closer to the answer. It's like a puzzle – each piece fits together to reveal the final picture.

Final Answer

The solution to the equation 64^{-3x-3} ullet 64 + 22 = 38 is:

$x = -8/9$

Conclusion

Woohoo! We've successfully solved the equation 64^{-3x-3} ullet 64 + 22 = 38. Remember, the key to solving exponential equations is to isolate the exponential term, simplify using exponent rules, express both sides with the same base, equate the exponents, and then solve for the variable. With practice, you'll become a pro at tackling these types of problems. Keep up the great work, and don't be afraid to explore more challenging equations. You've got this!

I hope this step-by-step guide has been helpful and has given you a solid understanding of how to solve exponential equations. Remember, math is like building a house – each concept is a brick, and as you lay more bricks, you build a stronger foundation. So, keep learning, keep practicing, and keep building your mathematical house!