Solving Exponential Equations Find The Value Of X

In the realm of mathematics, exponential equations often present a fascinating challenge, demanding a blend of algebraic manipulation and a solid grasp of exponent rules. This article delves into a specific exponential equation, providing a step-by-step solution and offering insights into the underlying mathematical principles. The equation we aim to solve is:

$8^{2x+3} \times \frac{1}{4^{3x}} = 2^{x+3}$

Our primary goal is to determine the value of x that satisfies this equation. To achieve this, we'll embark on a journey of simplification, employing the fundamental properties of exponents and logarithms. Let's begin by transforming the equation into a more manageable form, setting the stage for unraveling the value of x.

Transforming the Equation: A Foundation for Solution

To effectively tackle the exponential equation, the initial crucial step involves expressing all terms with a common base. In this instance, the base of 2 emerges as the ideal choice. We recognize that 8 can be expressed as $2^3$ and 4 as $2^2$. Substituting these equivalents into the original equation lays the groundwork for simplification. This transformation not only streamlines the equation but also brings us closer to isolating the variable x. The process of converting to a common base is a cornerstone technique in solving exponential equations, allowing us to manipulate exponents with greater ease. By rewriting the terms, we create a unified structure that facilitates the application of exponent rules, ultimately leading to the solution.

$\qquad 8^{2x+3} \times \frac{1}{4^{3x}} = 2^{x+3}$

$\qquad (2^3)^{2x+3} \times \frac{1}{(2^2)^{3x}} = 2^{x+3}$

The next step in our journey involves applying the power of a power rule, a fundamental concept in exponent manipulation. This rule dictates that when raising a power to another power, we multiply the exponents. Applying this principle to both the terms on the left side of the equation allows us to further simplify the expression. This simplification is a critical step in our quest to isolate x, as it consolidates the exponents and eliminates the parentheses. By applying the power of a power rule, we transform the equation into a more manageable form, paving the way for subsequent steps in the solution process. This strategic application of exponent rules demonstrates the interconnectedness of mathematical concepts and their role in solving complex problems.

$\qquad 2^{3(2x+3)} \times \frac{1}{2^{2(3x)}} = 2^{x+3}$

$\qquad 2^{6x+9} \times \frac{1}{2^{6x}} = 2^{x+3}$

The third step involves simplifying the fraction by recognizing that dividing by a power is equivalent to multiplying by the negative power of that base. This principle allows us to rewrite the fraction as a term with a negative exponent, further streamlining the equation. This transformation is crucial as it eliminates the fractional term, making the equation easier to manipulate. By expressing the division as multiplication with a negative exponent, we bring the equation closer to a form where we can directly apply the rules of exponent addition and subtraction. This step highlights the flexibility of exponent notation and its role in simplifying complex expressions.

$\qquad 2^{6x+9} \times 2^{-6x} = 2^{x+3}$

Simplifying and Solving for x: Unveiling the Solution

With the equation now in a more streamlined form, we can proceed to the next crucial step: applying the product of powers rule. This fundamental rule states that when multiplying powers with the same base, we add the exponents. Applying this rule to the left side of the equation allows us to combine the terms, further simplifying the expression. This simplification is a key step in our quest to isolate x, as it reduces the number of terms and brings us closer to a form where we can directly compare exponents. By applying the product of powers rule, we demonstrate the power of mathematical principles in simplifying complex equations and paving the way for a clear solution.

$\qquad 2^{(6x+9) + (-6x)} = 2^{x+3}$

$\qquad 2^9 = 2^{x+3}$

Now that both sides of the equation have the same base, we can leverage a fundamental property of exponential equations: if $a^m = a^n$, then m = n. This principle allows us to equate the exponents, transforming the exponential equation into a simple linear equation. This transformation is a critical step in our solution process, as it allows us to isolate x and determine its value. By equating the exponents, we bridge the gap between exponential expressions and linear equations, demonstrating the interconnectedness of mathematical concepts. This step highlights the power of mathematical reasoning in transforming complex problems into simpler, solvable forms.

$\qquad 9 = x + 3$

Finally, with the equation simplified to a linear form, solving for x becomes a straightforward algebraic exercise. By subtracting 3 from both sides of the equation, we isolate x and determine its value. This final step represents the culmination of our efforts, providing the solution to the original exponential equation. The simplicity of this step underscores the power of the preceding simplifications, demonstrating how a complex problem can be broken down into manageable components. The solution, x = 6, represents the value that satisfies the original equation, completing our mathematical journey.

$\qquad x = 9 - 3$

$\qquad x = 6$

Therefore, the value of x that satisfies the equation is 6. This corresponds to option C in the given choices.

Conclusion: A Journey Through Exponential Equations

In this exploration of an exponential equation, we've traversed a path of simplification, applying fundamental exponent rules and algebraic manipulations. From transforming the equation to a common base to equating exponents, each step has illuminated the underlying mathematical principles at play. The final solution, x = 6, not only answers the specific question but also reinforces the broader understanding of how exponential equations can be solved. This journey serves as a testament to the power of mathematical reasoning and the elegance of problem-solving techniques in the realm of exponents.

By mastering these techniques, you'll be well-equipped to tackle a wide range of exponential equations, confidently navigating the world of mathematical challenges.

Final Answer: The final answer is $\boxed{6}$