Solving Exponential Equations: A Step-by-Step Guide

Hey guys! Today, we're diving into the exciting world of exponential equations and tackling a problem that might seem a bit intimidating at first glance. But don't worry, we'll break it down step-by-step so you can conquer these equations like a pro. We're going to solve the equation (5 \sqrt{5})^{-2x+1} = \frac{1}{5} ullet 125^{x-3}. Trust me, by the end of this article, you'll feel confident in your ability to solve similar problems. So, let's get started!

Understanding Exponential Equations

Before we jump into the solution, let's quickly recap what exponential equations are all about. In essence, these are equations where the variable appears in the exponent. They often involve manipulating exponents and bases to find the unknown value of the variable. Solving these types of equations is a fundamental skill in mathematics, with applications ranging from compound interest calculations in finance to radioactive decay in physics. The core principle behind solving exponential equations lies in manipulating both sides of the equation to have the same base. Once the bases are identical, we can equate the exponents and solve the resulting algebraic equation. This process often involves using the laws of exponents to simplify expressions and rewrite numbers in different forms. For instance, expressing numbers as powers of a common base is a crucial technique. This involves identifying the prime factors of the numbers involved and rewriting them in exponential form. By understanding these fundamental concepts and techniques, you'll be well-equipped to tackle a wide variety of exponential equations and apply them in various real-world scenarios. Keep practicing, and you'll find that solving exponential equations becomes second nature.

Step 1: Expressing all terms with the same base

The key to solving this equation, and many exponential equations, is to express all the terms with the same base. Looking at our equation, (5 \sqrt{5})^{-2x+1} = \frac{1}{5} ullet 125^{x-3}, we can see that 5 is a common factor. So, let's try to rewrite everything in terms of base 5. This is a crucial step, guys, because it allows us to directly compare the exponents and simplify the equation. We need to remember the properties of exponents and how to manipulate them effectively. The first term, $(5 \sqrt{5})^{-2x+1}$, can be rewritten by recognizing that $\sqrt{5}$ is the same as $5^{1/2}$. This allows us to combine the terms inside the parentheses using the rule of exponents that states $a^m \cdot a^n = a^{m+n}$. The second term, $\frac{1}{5}$, can be easily expressed as $5^{-1}$. For the third term, $125^{x-3}$, we recognize that 125 is $5^3$, so we can rewrite it as $(5^3)^{x-3}$. By applying these transformations, we bring the equation closer to a form where the exponents can be equated and the equation can be solved. This initial step of expressing terms with a common base is the foundation for solving exponential equations effectively and accurately.

Let's break down each part:

• $\sqrt{5}$ can be written as $5^{\frac{1}{2}}$
• So, 5\sqrt{5} = 5^1 ullet 5^{\frac{1}{2}} = 5^{1+\frac{1}{2}} = 5^{\frac{3}{2}}
• $\frac{1}{5}$ can be written as $5^{-1}$
• $125$ can be written as $5^3$

Now, substitute these back into the original equation:

(5^{\frac{3}{2}})^{-2x+1} = 5^{-1} ullet (5^3)^{x-3}

Step 2: Simplify the equation using exponent rules

Now that we have everything in base 5, let's simplify using the rules of exponents. Remember, the power of a power rule states that $(a^m)^n = a^{mn}$. This rule is super important for simplifying exponential expressions and making the equation easier to solve. It allows us to multiply the exponents and reduce the complexity of the equation. Applying this rule, we can simplify terms like $(5^{\frac{3}{2}})^{-2x+1}$ and $(5^3)^{x-3}$ by multiplying the exponents. This step is crucial for transforming the equation into a form where we can directly compare the exponents and solve for the variable. By carefully applying the power of a power rule, we can eliminate the parentheses and combine the exponents, leading us closer to the solution. So, let's use this rule to simplify both sides of our equation and pave the way for the next steps in solving for x. Mastering these exponent rules is essential for handling exponential equations effectively.

Applying the power of a power rule:

5^{\frac{3}{2}(-2x+1)} = 5^{-1} ullet 5^{3(x-3)}

Simplify the exponents:

5^{-3x+\frac{3}{2}} = 5^{-1} ullet 5^{3x-9}

Now, use the product of powers rule, which states that a^m ullet a^n = a^{m+n}:

$5^{-3x+\frac{3}{2}} = 5^{-1+3x-9}$

$5^{-3x+\frac{3}{2}} = 5^{3x-10}$

Step 3: Equate the exponents

This is where the magic happens! Since the bases are the same (both are 5), we can equate the exponents. This is a fundamental technique in solving exponential equations, and it's what makes the whole process work. When the bases are identical, the equation essentially boils down to a simpler algebraic equation involving the exponents. This step allows us to eliminate the exponential form and focus on solving for the variable using basic algebraic techniques. By equating the exponents, we transform a potentially complex equation into a more manageable linear equation. This is a crucial step towards finding the solution, as it allows us to isolate the variable and determine its value. So, remember, when the bases match, equate the exponents – it's your key to solving exponential equations!

$-3x + \frac{3}{2} = 3x - 10$

Step 4: Solve for x

Alright, now we have a simple linear equation. Let's solve for x. This part involves using basic algebraic manipulations to isolate the variable x on one side of the equation. We'll be adding and subtracting terms, as well as multiplying or dividing to get x by itself. This is a standard process in algebra, and it's crucial for finding the numerical value of the unknown variable. By carefully performing these operations, we can systematically eliminate terms and eventually determine the value of x that satisfies the original equation. This step is the culmination of all the previous simplifications and transformations, and it provides the final answer to the problem. So, let's put our algebraic skills to the test and find the value of x that solves the exponential equation. Remember, precision and accuracy in these steps are key to arriving at the correct solution.

Add 3x to both sides:

$\frac{3}{2} = 6x - 10$

Add 10 to both sides:

$\frac{3}{2} + 10 = 6x$

$\frac{3}{2} + \frac{20}{2} = 6x$

$\frac{23}{2} = 6x$

Divide both sides by 6:

x = \frac{23}{2} ullet \frac{1}{6}

$x = \frac{23}{12}$

Step 5: Verify the Solution

It's always a good idea, guys, to verify your solution by plugging it back into the original equation. This step is crucial for ensuring that the value we found for x actually satisfies the equation. By substituting the solution back into the original equation, we can check if both sides of the equation are equal. This process helps us catch any potential errors we might have made during the simplification or solving steps. Verifying the solution provides confidence in our answer and confirms that we have correctly solved the exponential equation. It's a best practice to always include this step in your problem-solving routine, as it can save you from submitting an incorrect answer. So, let's take a moment to plug our solution back into the original equation and make sure everything checks out.

Let's plug $x = \frac{23}{12}$ back into the original equation:

(5 \sqrt{5})^{-2(\frac{23}{12})+1} = \frac{1}{5} ullet 125^{(\frac{23}{12})-3}

Simplify:

(5 \sqrt{5})^{-\frac{23}{6}+1} = \frac{1}{5} ullet 125^{\frac{23}{12}-\frac{36}{12}}

(5 \sqrt{5})^{-\frac{17}{6}} = \frac{1}{5} ullet 125^{-\frac{13}{12}}

Rewrite in terms of base 5:

(5^{\frac{3}{2}})^{-\frac{17}{6}} = 5^{-1} ullet (5^3)^{-\frac{13}{12}}

5^{-\frac{17}{4}} = 5^{-1} ullet 5^{-\frac{13}{4}}

$5^{-\frac{17}{4}} = 5^{-1-\frac{13}{4}}$

$5^{-\frac{17}{4}} = 5^{-\frac{17}{4}}$

The solution checks out!

Conclusion

So, the solution to the equation (5 \sqrt{5})^{-2x+1} = \frac{1}{5} ullet 125^{x-3} is $x = \frac{23}{12}$. We did it! Remember, the key to solving exponential equations is to express all terms with the same base, simplify using exponent rules, equate the exponents, and then solve for the variable. Don't forget to verify your solution to ensure accuracy. Keep practicing, and you'll master these types of problems in no time. You got this!