Solving Exponential Equations: A Step-by-Step Guide

Hey everyone! Let's dive into the world of exponential equations. These equations might look a bit intimidating at first glance, but trust me, they're totally manageable. We're going to break down the equation $(3 \sqrt{3})^{-x+1} = \frac{1}{3} \cdot 27^{x-5}$ step by step, making sure you understand every single move. No fancy math jargon, just clear explanations to help you conquer these problems. So, grab your pens and paper (or your digital tablets), and let's get started! The primary goal here is to isolate the variable, x, and find its value. To do this, we will leverage the power of exponential rules and logarithmic properties, simplifying each side until we can directly compare the exponents.

Understanding Exponential Equations: The Basics

Before we jump into the equation, let's make sure we're all on the same page with the basics. An exponential equation is simply an equation where the variable appears in the exponent. You know, things like $2^x = 8$ or $5^{2x-1} = 25$. The key to solving these equations is to get the same base on both sides. Why? Because if the bases are the same, then the exponents must be equal. It's like saying if $a^m = a^n$, then $m = n$. Easy peasy, right? We will be using the properties of exponents. For example, $a^{m} * a^{n} = a^{m+n}$ , $\frac{a^{m}}{a^{n}} = a^{m-n}$, $(a^{m})^{n} = a^{m*n}$, $a^{-m} = \frac{1}{a^{m}}$, and $a^{0} = 1$. These are the tools of the trade. Getting familiar with them is half the battle. Being able to spot when and how to use these rules is the key. Remember, the more you practice, the easier it gets. Don't worry if it seems a bit tricky at first; practice makes perfect. Let's get to the equation and break it down step-by-step, making sure you understand every single move. No fancy math jargon, just clear explanations to help you conquer these problems. So, grab your pens and paper (or your digital tablets), and let's get started! To solve for x, we want to rewrite each side of the equation with the same base. This will allow us to equate the exponents and solve for x. It's a clever trick, but it simplifies the equation significantly. The core concept is this: by expressing both sides with the same base, we transform an exponential equation into a much simpler algebraic equation. Let's do this!

We will use these basic rules and the properties of exponents to transform the given equation step-by-step.

Step-by-Step Solution: Breaking Down the Equation

Alright, let's tackle that equation: $(3 \sqrt{3})^{-x+1} = \frac{1}{3} \cdot 27^{x-5}$. Our goal is to rewrite both sides with the same base. It's like we are looking for a common language that both sides of the equation can understand. We will now transform the terms to have the same base. First, let's simplify $3\sqrt{3}$. Remember that $\sqrt{3} = 3^{\frac{1}{2}}$. So, we have $3 \cdot 3^{\frac{1}{2}}$. Then, using the rule $a^{m} * a^{n} = a^{m+n}$, we get $3^{1 + \frac{1}{2}} = 3^{\frac{3}{2}}$. Now, our equation becomes $(3^{\frac{3}{2}})^{-x+1} = \frac{1}{3} \cdot 27^{x-5}$. Cool, right? Next, let's deal with the right side of the equation. We know that $27 = 3^3$. So, we can rewrite the right side as $\frac{1}{3} \cdot (3^3)^{x-5}$. Using the power of a power rule, $(a^{m})^{n} = a^{m*n}$, we get $\frac{1}{3} \cdot 3^{3(x-5)}$. Remember that $\frac{1}{3} = 3^{-1}$. Now, our equation is $3^{\frac{3}{2}(-x+1)} = 3^{-1} \cdot 3^{3(x-5)}$. Using the rule, $a^{m} * a^{n} = a^{m+n}$, we have $3^{\frac{3}{2}(-x+1)} = 3^{-1 + 3(x-5)}$. This simplifies to $3^{\frac{3}{2}(-x+1)} = 3^{-1 + 3x - 15}$, which is $3^{\frac{3}{2}(-x+1)} = 3^{3x - 16}$. Now, we have both sides with the same base, which is 3. According to the exponential rule, when the bases are the same, then the exponents must be equal. Hence, we can set the exponents equal to each other to solve for x. This transforms the problem into an algebraic equation, which is much easier to solve. Let's do this and solve for the variable x.

Simplifying Exponents

Now that we have the same base on both sides, we can equate the exponents: $\frac{3}{2}(-x + 1) = 3x - 16$. Expanding the left side gives us $-\frac{3}{2}x + \frac{3}{2} = 3x - 16$. We will want to isolate the variable, x. Add $\frac{3}{2}x$ to both sides: $\frac{3}{2} = 3x + \frac{3}{2}x - 16$. Simplify and add 16 to both sides: $16 + \frac{3}{2} = 3x + \frac{3}{2}x$. Now, we have $\frac{35}{2} = \frac{9}{2}x$. To solve for x, we multiply both sides by $\frac{2}{9}$: $x = \frac{35}{2} \cdot \frac{2}{9}$. The 2's cancel each other, and we are left with $x = \frac{35}{9}$.

Therefore, the value of x that satisfies the original equation is $\frac{35}{9}$. Pretty neat, huh? Always remember to double-check your answers! Plugging the value back into the original equation ensures we haven't made any mistakes. Let's go over this example again, so you get a better idea of how to work with exponential equations.

Double-Checking Our Work

Always a good idea, guys, to double-check your answer. Let's plug $x = \frac{35}{9}$ back into the original equation to make sure we did everything correctly. Our original equation was $(3 \sqrt{3})^{-x+1} = \frac{1}{3} \cdot 27^{x-5}$. Substituting $x = \frac{35}{9}$ we get:

$(3 \sqrt{3})^{-\frac{35}{9} + 1} = \frac{1}{3} \cdot 27^{\frac{35}{9} - 5}$

Simplify the exponents. Remember $3 \sqrt{3} = 3^{\frac{3}{2}}$. So, the left side becomes:

$(3^{\frac{3}{2}})^{-\frac{26}{9}} = (3^{\frac{3}{2}})^{-\frac{26}{9}} = 3^{-\frac{26}{6}} = 3^{-\frac{13}{3}}$

For the right side of the equation we get:

$\frac{1}{3} \cdot 27^{\frac{-10}{9}} = 3^{-1} \cdot (3^3)^{-\frac{10}{9}} = 3^{-1} \cdot 3^{-\frac{30}{9}} = 3^{-1} \cdot 3^{-\frac{10}{3}} = 3^{-\frac{3}{3}} \cdot 3^{-\frac{10}{3}} = 3^{-\frac{13}{3}}$

Both sides equal each other, therefore, the solution checks out!

Tips and Tricks for Solving Exponential Equations

• Practice, practice, practice! The more you solve these equations, the more comfortable you'll become. Try different problems, and don't be afraid to make mistakes. That's how we learn, right? Keep practicing, and you'll become a master of these equations! Solve as many problems as you can! It’s the best way to improve. Familiarize yourself with the various forms in which exponential equations can be presented. This way, you’ll be prepared to recognize the patterns and apply the correct methods. Don't be afraid to seek help. Sometimes, a different perspective or a quick hint can make all the difference.
• Know your exponent rules! The rules of exponents are your best friends here. Make sure you're comfortable with them. Review the rules, such as how to handle negative exponents, fractional exponents, and what happens when you raise a power to another power. They're the foundation for simplifying the equations.
• Always look for common bases. This is your first step. Try to rewrite the equation so both sides have the same base. If you can't find an obvious common base, consider using logarithms.
• Simplify, simplify, simplify! Reduce everything to its simplest form. This means simplifying the exponents, combining like terms, and isolating the variable.
• Double-check your work! Always, always plug your answer back into the original equation to make sure it works. It’s easy to make a small mistake somewhere along the way. Checking your answer helps you catch any errors before you move on. It also builds your confidence and makes sure you've correctly solved the problem.
• Don't give up! Some equations might seem tough, but don't get discouraged. Take your time, break down the problem, and keep working through it. You've got this!

Conclusion: Mastering Exponential Equations

So, there you have it! We've successfully solved an exponential equation. We started with a seemingly complex problem and, using our knowledge of exponents, we simplified it step by step. Remember the key takeaways: Find a common base, use the exponent rules, and simplify until you isolate the variable. With practice and patience, you'll be able to tackle any exponential equation that comes your way. Exponential equations are a fundamental concept in mathematics, with applications in various fields such as finance, physics, and computer science. They are used to model growth, decay, and other dynamic processes. Therefore, understanding and mastering exponential equations is very important! I hope this guide has been helpful. Keep practicing, keep learning, and you'll do great! Let me know if you have any questions in the comments below. Thanks for reading, and happy solving!