Solving (1/4)^(3z-1) = 16^(z+2) * 64^(z-2) A Step-by-Step Guide

Jul 9, 2025 by ADMIN 64 views

Solving Exponential Equations A Comprehensive Guide to Finding 'z'

In the realm of mathematics, exponential equations hold a significant place, often presenting intriguing challenges that require a solid understanding of exponent rules and algebraic manipulation. This article delves into the process of solving the exponential equation (1/4)^(3z-1) = 16^(z+2) * 64^(z-2), providing a step-by-step guide that not only unravels the solution but also reinforces key concepts related to exponents and equation-solving techniques. Our main goal is to solve for the unknown variable 'z', and in doing so, we will explore various properties of exponents, including the power of a power rule, the product of powers rule, and the quotient of powers rule. By the end of this comprehensive guide, you will not only have the solution to this specific equation but also a broader understanding of how to approach similar exponential problems.

Exponential equations are equations in which the variable appears in the exponent. These equations are crucial in various fields such as physics, engineering, and finance, where exponential growth and decay models are frequently used. To effectively tackle exponential equations, it is essential to grasp the fundamental principles of exponents and logarithms. The equation we are about to solve serves as an excellent example to illustrate these principles and techniques.

In this article, we will break down the equation into manageable parts, systematically applying exponent rules to simplify and ultimately solve for 'z.' We will start by expressing all terms with a common base, a crucial step in solving exponential equations. This involves recognizing that 1/4, 16, and 64 can all be written as powers of 2. By doing so, we can create a uniform base across the equation, making it easier to equate the exponents. The power of a power rule will then come into play, allowing us to simplify expressions where an exponent is raised to another exponent. Furthermore, the product of powers rule will help us combine terms on one side of the equation, ultimately leading us to a linear equation in 'z.' Solving this linear equation will give us the value of 'z' that satisfies the original exponential equation.

Throughout this guide, we will emphasize clarity and precision in our steps. Each transformation of the equation will be explained in detail, ensuring that you understand not just the 'how' but also the 'why' behind each step. This approach is designed to foster a deeper understanding of exponential equations and problem-solving strategies. Whether you are a student learning algebra or someone looking to refresh your math skills, this article offers a valuable resource for mastering exponential equations. So, let's dive in and unravel the intricacies of this equation together.

Step 1: Express All Terms with a Common Base

The first crucial step in solving the exponential equation

$\\\left(\\\\frac{1}{4}\\\\\right)^{3 z-1}=16^{z+2} \\\\cdot 64^{z-2}$

involves expressing all terms with a common base. This simplifies the equation and allows us to equate the exponents. Recognizing that 1/4, 16, and 64 can all be expressed as powers of 2 is the key to this step. This transformation is essential because it creates a uniform foundation for the equation, making it easier to manipulate and solve. By expressing all terms with the same base, we can apply the properties of exponents more effectively and reduce the equation to a more manageable form.

Let's begin by rewriting each term individually. We know that 1/4 is the reciprocal of 4, and 4 is 2 squared (2^2). Therefore, 1/4 can be written as 2^(-2). The exponent (3z - 1) remains unchanged for now, so the left side of the equation becomes (2^(-2))(3z - 1). This transformation is a critical application of the definition of negative exponents and the relationship between fractions and powers.

Next, we consider the term 16^(z + 2). We recognize that 16 is 2 to the power of 4 (2^4). Thus, 16^(z + 2) can be rewritten as (2⁴⁾(z + 2). This step is a straightforward application of recognizing powers of a common base, which is a fundamental skill in solving exponential equations. By expressing 16 as a power of 2, we prepare it for further simplification using the power of a power rule.

Finally, we address the term 64^(z - 2). Similarly, we recognize that 64 is 2 to the power of 6 (2^6). Therefore, 64^(z - 2) can be rewritten as (2⁶⁾(z - 2). This conversion completes the process of expressing all numerical bases in terms of the common base 2. Now, each term in the equation is expressed as a power of 2, setting the stage for the next phase of simplification.

By rewriting the equation in terms of the common base 2, we have successfully transformed the original equation into a form that is much easier to work with. The equation now looks like this: (2^(-2))(3z - 1) = (2⁴⁾(z + 2) * (2⁶⁾(z - 2). This transformation is a cornerstone technique in solving exponential equations, as it allows us to apply exponent rules and simplify the equation to a linear form. The subsequent steps will involve applying these exponent rules to further simplify and solve for 'z.' The importance of this initial step cannot be overstated, as it lays the groundwork for the rest of the solution process. Understanding how to identify and use common bases is crucial for solving a wide range of exponential equations.

Step 2: Apply the Power of a Power Rule

Now that we have expressed all terms with a common base, which is 2, the next step in solving the equation

$\\\left(\\\\frac{1}{4}\\\\\right)^{3 z-1}=16^{z+2} \\\\cdot 64^{z-2}$

involves applying the power of a power rule. This rule states that (a^m)n = a^(m*n), where 'a' is the base and 'm' and 'n' are exponents. This rule is a cornerstone of exponent manipulation and is essential for simplifying expressions where an exponent is raised to another exponent. Applying this rule will help us eliminate the parentheses in our equation and consolidate the exponents.

Let's start with the left side of the equation: (2^(-2))(3z - 1). Applying the power of a power rule, we multiply the exponents -2 and (3z - 1), resulting in 2^(-2 * (3z - 1)). Distributing the -2 gives us 2^(-6z + 2). This simplification significantly reduces the complexity of the left side of the equation, making it easier to work with in the subsequent steps.

Next, we apply the power of a power rule to the first term on the right side of the equation: (2⁴⁾(z + 2). Multiplying the exponents 4 and (z + 2), we get 2^(4 * (z + 2)). Distributing the 4 gives us 2^(4z + 8). Again, this application of the power of a power rule simplifies the expression and brings us closer to isolating the variable 'z'.

Finally, we apply the power of a power rule to the second term on the right side of the equation: (2⁶⁾(z - 2). Multiplying the exponents 6 and (z - 2), we obtain 2^(6 * (z - 2)). Distributing the 6 gives us 2^(6z - 12). This completes the application of the power of a power rule to all terms in the equation.

After applying the power of a power rule, our equation now looks like this: 2^(-6z + 2) = 2^(4z + 8) * 2^(6z - 12). This transformed equation is much simpler than the original, with all exponents now expressed as linear expressions in 'z'. The next step will involve using the product of powers rule to combine the terms on the right side of the equation. By systematically applying these exponent rules, we are gradually transforming the equation into a form that is easily solvable.

The power of a power rule is a fundamental tool in algebra, especially when dealing with exponential equations. Its proper application is crucial for simplifying complex expressions and revealing the underlying structure of the equation. Understanding and mastering this rule is essential for anyone looking to excel in algebra and related fields. In the following steps, we will continue to leverage these rules to isolate and solve for the unknown variable 'z'.

Step 3: Apply the Product of Powers Rule

With the power of a power rule successfully applied, the equation now stands as 2^(-6z + 2) = 2^(4z + 8) * 2^(6z - 12). The next step in solving this exponential equation involves applying the product of powers rule. This rule states that a^m * a^n = a^(m + n), where 'a' is the base and 'm' and 'n' are exponents. This rule is crucial for simplifying expressions where terms with the same base are multiplied. By applying this rule, we can combine the two terms on the right side of the equation into a single term, further simplifying the equation.

Looking at the right side of the equation, we have 2^(4z + 8) * 2^(6z - 12). According to the product of powers rule, we can combine these terms by adding their exponents. Thus, we have 2^((4z + 8) + (6z - 12)). Adding the exponents, we get 4z + 8 + 6z - 12, which simplifies to 10z - 4. Therefore, the right side of the equation becomes 2^(10z - 4).

Now, our equation looks like this: 2^(-6z + 2) = 2^(10z - 4). This equation is significantly simpler than the original. Both sides of the equation now have the same base (2), and each side is expressed as a single exponential term. This simplification is a direct result of the product of powers rule, which allowed us to combine multiple terms into one.

The product of powers rule is a fundamental concept in algebra and is widely used in various mathematical contexts. Understanding how to apply this rule is essential for simplifying expressions and solving equations. In the context of exponential equations, the product of powers rule is particularly useful for combining terms and reducing the complexity of the equation.

By applying the product of powers rule, we have transformed the equation into a form where both sides have the same base. This sets the stage for the next critical step: equating the exponents. Since the bases are the same, for the equation to hold true, the exponents must be equal. This principle will allow us to transition from an exponential equation to a linear equation, which is much easier to solve. In the subsequent steps, we will focus on equating the exponents and solving the resulting linear equation for 'z'. The systematic application of exponent rules has brought us closer to the final solution.

Step 4: Equate the Exponents

With the equation simplified to 2^(-6z + 2) = 2^(10z - 4), we have reached a crucial juncture in solving for 'z'. The next step involves equating the exponents. This step is based on the fundamental principle that if a^m = a^n, then m = n, provided that 'a' is a positive number not equal to 1. This principle is a direct consequence of the one-to-one property of exponential functions and is the key to converting an exponential equation into a linear equation.

In our equation, the base is 2, which satisfies the condition for equating exponents. Therefore, we can set the exponents equal to each other: -6z + 2 = 10z - 4. This transformation is a pivotal moment in the solution process, as it converts the exponential equation into a linear equation, which is far easier to solve. The equation -6z + 2 = 10z - 4 is a linear equation in one variable, 'z', and can be solved using basic algebraic techniques.

The process of equating exponents is a powerful tool in solving exponential equations. It allows us to bypass the complexities of exponential expressions and focus on the linear relationships between the exponents. This step is not only mathematically sound but also simplifies the problem significantly, making it accessible to a wider range of problem-solving approaches.

The equation -6z + 2 = 10z - 4 now represents a straightforward algebraic problem. To solve for 'z', we will need to isolate the variable on one side of the equation. This involves adding and subtracting terms to both sides of the equation to group the 'z' terms together and the constant terms together. The subsequent steps will focus on these algebraic manipulations to ultimately find the value of 'z' that satisfies the original exponential equation.

By equating the exponents, we have effectively bridged the gap between exponential and linear equations. This transition is a testament to the power of mathematical principles in simplifying complex problems. The following steps will involve applying basic algebraic techniques to solve the resulting linear equation, bringing us closer to the final solution for 'z'. The ability to recognize and apply this principle is crucial for mastering exponential equations and related mathematical concepts.

Step 5: Solve the Linear Equation for 'z'

Having equated the exponents, we now have the linear equation -6z + 2 = 10z - 4. The final step in solving for 'z' involves manipulating this equation to isolate 'z' on one side. This process requires applying basic algebraic principles, such as adding and subtracting terms from both sides of the equation to maintain equality. The goal is to group the 'z' terms on one side and the constant terms on the other, making it straightforward to solve for 'z'.

First, let's add 6z to both sides of the equation to move the 'z' term from the left side to the right side. This gives us: -6z + 2 + 6z = 10z - 4 + 6z, which simplifies to 2 = 16z - 4. Adding the same term to both sides of the equation ensures that the equation remains balanced and the equality is preserved.

Next, we want to isolate the 'z' term further, so we add 4 to both sides of the equation. This gives us: 2 + 4 = 16z - 4 + 4, which simplifies to 6 = 16z. This step moves the constant term from the right side to the left side, bringing us closer to isolating 'z'.

Now, we have the equation 6 = 16z. To solve for 'z', we need to divide both sides of the equation by 16. This gives us: 6 / 16 = 16z / 16, which simplifies to z = 6/16. This division isolates 'z' and provides us with its value.

Finally, we can simplify the fraction 6/16 by dividing both the numerator and the denominator by their greatest common divisor, which is 2. This gives us z = 3/8. Therefore, the solution to the linear equation, and consequently the original exponential equation, is z = 3/8.

Solving linear equations is a fundamental skill in algebra, and this step demonstrates the importance of mastering these basic techniques. The process of adding, subtracting, multiplying, and dividing terms to isolate the variable is a core concept in mathematics and is applicable in a wide range of problem-solving scenarios.

By systematically applying algebraic principles, we have successfully solved the linear equation and found the value of 'z'. This final step completes the solution to the original exponential equation, demonstrating the power of combining exponent rules and algebraic manipulation techniques. The solution z = 3/8 satisfies the original equation, and we have arrived at this solution through a series of logical and well-defined steps. The journey from the initial exponential equation to the final solution highlights the interconnectedness of mathematical concepts and the importance of a step-by-step approach to problem-solving.

Solution

After meticulously working through the steps, we have arrived at the solution for the exponential equation

$\\\left(\\\\frac{1}{4}\\\\\right)^{3 z-1}=16^{z+2} \\\\cdot 64^{z-2}$ .

The value of 'z' that satisfies this equation is z = 3/8. This solution is the culmination of several key steps, including expressing all terms with a common base, applying the power of a power rule, applying the product of powers rule, equating the exponents, and finally, solving the resulting linear equation. Each step was crucial in transforming the original complex exponential equation into a manageable linear equation, which we then solved using basic algebraic techniques.

The process of solving this equation has reinforced several important concepts in algebra, particularly those related to exponents and equation-solving strategies. The ability to recognize and apply exponent rules, such as the power of a power rule and the product of powers rule, is essential for simplifying exponential expressions. Furthermore, the principle of equating exponents when the bases are the same is a powerful tool for converting exponential equations into linear equations. These concepts are fundamental in various areas of mathematics and are widely used in scientific and engineering applications.

The solution z = 3/8 represents the point at which the two sides of the original equation are equal. Substituting this value back into the original equation would verify that both sides indeed have the same value. This process of verification is a good practice to ensure the accuracy of the solution and to reinforce understanding of the equation.

In summary, the solution z = 3/8 is the answer to the exponential equation that we set out to solve. The journey to this solution has been a comprehensive exploration of exponent rules, algebraic manipulation, and problem-solving techniques. This exercise not only provides the answer to a specific problem but also enhances our understanding of the broader mathematical principles involved. The ability to solve exponential equations is a valuable skill, and this detailed walkthrough serves as a guide for tackling similar problems in the future.