Solve The Equation (1/4)^(3z-1) = 16^(z+2) * 64^(z-2) For Z.

In the realm of mathematics, exponential equations hold a unique position, often presenting both a challenge and an opportunity for insightful problem-solving. These equations, characterized by variables nestled within exponents, require a specific set of techniques to unravel their solutions. This article delves into the process of solving one such exponential equation, providing a comprehensive, step-by-step guide that illuminates the underlying principles and empowers you to tackle similar problems with confidence. Our focus will be on the equation $(\frac{1}{4})^{3z-1}=16^{z+2} \cdot 64^{z-2}$, where our primary goal is to determine the elusive value of 'z'. To embark on this mathematical journey, we'll leverage the power of exponent rules, strategic simplification, and a touch of algebraic manipulation. By the end of this exploration, you'll not only possess the solution to this particular equation but also gain a deeper understanding of the methodologies employed in solving exponential equations in general.

Unveiling the Foundation: Exponent Rules

Before we dive into the nitty-gritty of solving our equation, it's crucial to establish a firm grasp of the exponent rules that will serve as our guiding principles. These rules, like the fundamental laws of nature, dictate how exponents behave and interact. Among the most relevant rules for our endeavor are:

1. The Power of a Power Rule: $(a^m)^n = a^{m \cdot n}$. This rule states that when raising a power to another power, we multiply the exponents.
2. The Product of Powers Rule: $a^m \cdot a^n = a^{m+n}$. This rule dictates that when multiplying powers with the same base, we add the exponents.
3. The Quotient of Powers Rule: $\frac{a^m}{a^n} = a^{m-n}$. This rule mirrors the product rule, stating that when dividing powers with the same base, we subtract the exponents.
4. The Negative Exponent Rule: $a^{-n} = \frac{1}{a^n}$. This rule provides a bridge between positive and negative exponents, allowing us to express reciprocals as powers.

These exponent rules, seemingly abstract at first glance, are the bedrock upon which our solution will be built. They provide the tools to manipulate and simplify exponential expressions, ultimately leading us to the value of 'z'. With these rules firmly in our grasp, we can now embark on the journey of solving our equation.

The Quest for a Common Base: Simplifying the Equation

At the heart of solving exponential equations lies the strategy of achieving a common base. This pivotal step allows us to directly compare exponents and establish a pathway to solving for the unknown variable. In our equation, $(\frac{1}{4})^{3z-1}=16^{z+2} \cdot 64^{z-2}$, the bases appear disparate: $\frac{1}{4}$, 16, and 64. However, a closer examination reveals that all these bases can be expressed as powers of 2, or more conveniently, powers of 4. This realization is the key to unlocking the equation's solution.

Let's embark on the transformation. We can rewrite $\frac{1}{4}$ as $4^{-1}$, since the negative exponent rule dictates that $a^{-n} = \frac{1}{a^n}$. Similarly, 16 can be expressed as $4^2$, and 64 can be represented as $4^3$. Armed with these equivalencies, we can rewrite our original equation as follows:

$(4^{-1})^{3z-1} = (4^2)^{z+2} \cdot (4^3)^{z-2}$

This seemingly simple substitution is a monumental step forward. By expressing all bases as powers of 4, we've paved the way for applying the power of a power rule, which will further simplify the equation and bring us closer to our goal.

Unleashing the Power of a Power: Applying Exponent Rules

With our equation now expressed in terms of a common base, we can unleash the power of the exponent rules, specifically the power of a power rule, which states that $(a^m)^n = a^{m \cdot n}$. This rule allows us to simplify expressions where a power is raised to another power, effectively multiplying the exponents.

Applying this rule to our equation, $(4^{-1})^{3z-1} = (4^2)^{z+2} \cdot (4^3)^{z-2}$, we obtain:

$4^{-1(3z-1)} = 4^{2(z+2)} \cdot 4^{3(z-2)}$

Simplifying the exponents, we get:

$4^{-3z+1} = 4^{2z+4} \cdot 4^{3z-6}$

Now, we encounter a product of powers on the right side of the equation. This is where the product of powers rule comes into play. This rule, which states that $a^m \cdot a^n = a^{m+n}$, allows us to combine the terms on the right side by adding their exponents.

Applying the product of powers rule, we have:

$4^{-3z+1} = 4^{(2z+4)+(3z-6)}$

Simplifying the exponent on the right side, we arrive at:

$4^{-3z+1} = 4^{5z-2}$

This is a pivotal moment in our solution. We've successfully transformed the equation into a form where both sides have the same base raised to different exponents. This allows us to equate the exponents and solve for 'z'.

Equating Exponents: Solving for z

Having maneuvered our equation to the form $4^{-3z+1} = 4^{5z-2}$, we've reached a crucial juncture. The beauty of this form lies in the fact that if two powers with the same base are equal, then their exponents must also be equal. This fundamental principle allows us to equate the exponents and create a simple algebraic equation that we can readily solve for 'z'.

Equating the exponents, we obtain:

$-3z + 1 = 5z - 2$

Now, we have a linear equation in one variable, a familiar territory for algebraic manipulation. To isolate 'z', we can add 3z to both sides of the equation:

$1 = 8z - 2$

Next, we add 2 to both sides:

$3 = 8z$

Finally, we divide both sides by 8 to solve for 'z':

$z = \frac{3}{8}$

And there we have it! The value of 'z' that satisfies the original exponential equation is $\frac{3}{8}$. This elegant solution is the culmination of our strategic simplification, application of exponent rules, and algebraic manipulation. With this value in hand, we can now take a moment to reflect on the journey we've undertaken.

The Journey's End: Reflecting on the Solution

Having successfully navigated the intricacies of the exponential equation, we arrive at the solution: $z = \frac{3}{8}$. This value, seemingly simple in its fractional form, represents the culmination of our mathematical efforts. But beyond the numerical answer, it's the journey of problem-solving that holds the true value. We've witnessed the power of strategic simplification, the elegance of exponent rules, and the precision of algebraic manipulation. These tools, honed through practice and understanding, are the keys to unlocking a world of mathematical challenges.

Throughout this exploration, we've not only solved a specific equation but also gained a deeper appreciation for the general principles of solving exponential equations. The strategy of achieving a common base, the judicious application of exponent rules, and the careful manipulation of algebraic expressions are all techniques that can be applied to a wide range of similar problems. This understanding empowers us to approach new challenges with confidence and a well-equipped mathematical toolkit.

So, as we conclude this exploration, let's carry forward the lessons learned, the strategies employed, and the satisfaction of a problem well-solved. The world of mathematics awaits, and with our newfound knowledge, we are ready to embrace its challenges and unlock its hidden beauty.

Conclusion

In conclusion, we have successfully solved the exponential equation $(\frac{1}{4})^{3z-1}=16^{z+2} \cdot 64^{z-2}$ by employing a series of strategic steps. We began by establishing a firm understanding of exponent rules, which served as our guiding principles. We then transformed the equation by expressing all bases as powers of 4, achieving a common base that allowed for further simplification. The power of a power rule and the product of powers rule were instrumental in reducing the equation to a manageable form. Finally, by equating exponents and solving the resulting linear equation, we arrived at the solution: $z = \frac{3}{8}$. This journey highlights the importance of strategic problem-solving, the power of mathematical tools, and the satisfaction of arriving at a solution through logical and systematic steps.