Solving Exponential Equations A Step-by-Step Guide

Introduction

In the realm of mathematics, solving exponential equations is a fundamental skill. Exponential equations, characterized by having the variable in the exponent, often appear in various mathematical contexts and real-world applications, such as population growth, compound interest, and radioactive decay. This comprehensive guide provides a step-by-step approach to solving the exponential equation 64^{-3x-3} ullet 64 + 22 = 38, complete with detailed explanations and justifications for each step. By mastering these techniques, you will enhance your ability to solve similar exponential problems and deepen your understanding of exponential functions.

Understanding Exponential Equations

Before diving into the solution, it's essential to understand the basic principles of exponential equations. An exponential equation is one in which the variable appears in the exponent. These equations are solved using the properties of exponents and logarithms. The key to solving these equations lies in manipulating the equation to isolate the exponential term and then applying appropriate logarithmic properties to solve for the variable. We will utilize these principles to dissect and conquer the given equation systematically. Exponential equations are a cornerstone of mathematical analysis and are crucial for modeling phenomena that exhibit exponential growth or decay. Proficiency in solving these equations is invaluable for anyone studying mathematics or related fields.

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

Let's address the problem at hand: solve the exponential equation 64^{-3x-3} ullet 64 + 22 = 38. This equation features the variable $x$ in the exponent, making it an exponential equation. Our goal is to isolate $x$ by systematically applying algebraic manipulations and the properties of exponents. The equation appears complex at first glance, but by breaking it down into manageable steps, we can solve it effectively. This problem is an excellent example of how exponential equations can be solved using a combination of algebraic techniques and an understanding of exponent rules. Successfully navigating this problem will equip you with the skills to tackle similar challenges in the future.

Step-by-Step Solution

1. Isolate the Exponential Term

Our first step is to isolate the exponential term in the equation. The given equation is 64^{-3x-3} ullet 64 + 22 = 38. To isolate the exponential term, we subtract 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

This step is crucial because it sets the stage for applying exponent rules and ultimately solving for $x$. By isolating the exponential term, we've simplified the equation and made it more amenable to further manipulation. This is a common strategy in solving exponential equations: get the exponential part alone on one side of the equation.

2. Simplify the Exponential Expression

Next, we simplify the exponential expression. We notice that $64$ can be written as $64^1$, so we have 64^{-3x-3} ullet 64^1 = 16. Now we can use the property of exponents that states a^m ullet a^n = a^{m+n}. Applying this property, we get:

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

Simplifying the exponent, we have:

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

This step condenses the exponential terms into a single expression, making the equation easier to handle. Using exponent properties is a key technique in solving exponential equations. By simplifying the expression, we move closer to a form where we can apply logarithms or further algebraic manipulation.

3. Express Both Sides with the Same Base

To solve for $x$, we need to express both sides of the equation with the same base. We can write both 64 and 16 as powers of 4: $64 = 4^3$ and $16 = 4^2$. Substituting these into our equation gives:

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

Now, we use the power of a power property, which states $(a^m)^n = a^{mn}$. Applying this property, we get:

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

Simplifying the exponent, we have:

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

This is a critical step because when the bases are the same, we can equate the exponents. This simplifies the equation significantly and allows us to solve for $x$ directly. Expressing both sides with the same base is a common strategy in solving exponential equations, enabling us to transform the problem into a more straightforward algebraic equation.

4. Equate the Exponents

Since the bases are the same, we can now equate the exponents. We have $4^{-9x-6} = 4^2$, so we can set the exponents equal to each other:

$-9x - 6 = 2$

This step transforms the exponential equation into a linear equation, which is much easier to solve. Equating the exponents is a direct consequence of the one-to-one property of exponential functions, which states that if $a^m = a^n$, then $m = n$. This principle is fundamental in solving exponential equations and is applied after ensuring that both sides of the equation have the same base.

5. Solve for $x$

Now we solve the linear equation for $x$. We have $-9x - 6 = 2$. First, add 6 to both sides:

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

$-9x = 8$

Next, divide both sides by -9:

x = -rac{8}{9}

Therefore, the solution to the equation is x = -rac{8}{9}. This step completes the algebraic process of solving for the variable. By systematically isolating $x$, we have arrived at the solution to the exponential equation. The solution x = -rac{8}{9} satisfies the original equation, as can be verified by substituting it back into the initial expression.

Answer

The correct answer is B. x = -rac{8}{9}. We have shown through a step-by-step solution how to arrive at this answer by isolating the exponential term, simplifying the expression, expressing both sides with the same base, equating the exponents, and solving for $x$. This comprehensive approach provides a clear and understandable method for tackling exponential equations. The process of solving exponential equations involves a combination of algebraic manipulation and understanding of exponent properties, making it a valuable skill in mathematics.

Conclusion

In conclusion, solving exponential equations requires a systematic approach, involving algebraic manipulation and the application of exponent properties. By isolating the exponential term, simplifying expressions, expressing both sides with the same base, equating the exponents, and solving for the variable, we can effectively solve equations like 64^{-3x-3} ullet 64 + 22 = 38. This guide has provided a detailed walkthrough of the solution, reinforcing the key concepts and techniques involved in solving exponential equations. Mastering these techniques will undoubtedly strengthen your mathematical abilities and problem-solving skills. Remember, practice is crucial for mastering these concepts, so continue to challenge yourself with similar problems to solidify your understanding.