Solving For X In (3/4)³ × (4/3)⁻⁷ = (9/16)ˣ A Step-by-Step Guide
Introduction
In this comprehensive guide, we will delve into the process of solving for x in the equation (3/4)³ × (4/3)⁻⁷ = (9/16)ˣ. This equation involves exponents and fractions, requiring a solid understanding of exponent rules and algebraic manipulation. Whether you're a student tackling algebra problems or simply looking to enhance your mathematical skills, this step-by-step solution will provide you with the necessary tools and insights. We'll break down each step, explaining the underlying principles and techniques, ensuring that you not only arrive at the correct answer but also grasp the fundamental concepts involved. By the end of this guide, you'll be well-equipped to tackle similar problems with confidence and precision. This exploration is essential for anyone seeking to master algebraic equations and exponent manipulation. Understanding these concepts is crucial for various mathematical applications and problem-solving scenarios. So, let's embark on this mathematical journey together and unlock the solution to this intriguing equation.
Understanding the Basics of Exponents
Before we dive into solving the equation, it's crucial to understand the fundamental principles of exponents. Exponents, also known as powers, indicate how many times a number (the base) is multiplied by itself. For instance, in the expression aⁿ, a is the base, and n is the exponent. This means a is multiplied by itself n times. A solid grasp of exponent rules is essential for simplifying and solving equations like the one we're tackling today. These rules allow us to manipulate expressions, making them easier to work with and ultimately leading to a solution.
One of the most important rules is the product of powers rule, which states that when multiplying powers with the same base, you add the exponents: aᵐ × aⁿ = aᵐ⁺ⁿ. Conversely, the quotient of powers rule states that when dividing powers with the same base, you subtract the exponents: aᵐ / aⁿ = aᵐ⁻ⁿ. Another critical rule is the power of a power rule, which says that when raising a power to another power, you multiply the exponents: (aᵐ)ⁿ = aᵐⁿ. Additionally, a negative exponent indicates a reciprocal: a⁻ⁿ = 1/aⁿ. This rule is particularly important in our equation. Finally, any non-zero number raised to the power of 0 is 1: a⁰ = 1. These rules are the building blocks for manipulating exponential expressions and are crucial for solving complex equations. Mastering these rules will significantly enhance your ability to simplify expressions and solve for unknowns, not just in this particular equation, but in a wide range of mathematical problems. In the following sections, we'll apply these rules to simplify the given equation and isolate the variable x.
Step 1: Simplify the Negative Exponent
The first step in solving the equation (3/4)³ × (4/3)⁻⁷ = (9/16)ˣ is to address the negative exponent. Recall that a negative exponent indicates a reciprocal. Specifically, a⁻ⁿ = 1/aⁿ. Applying this rule to our equation, we have (4/3)⁻⁷, which can be rewritten as (3/4)⁷. This transformation is crucial because it allows us to work with positive exponents, which are generally easier to manipulate. By converting the negative exponent to a positive one, we simplify the expression and make it more manageable for further calculations. This step not only makes the equation more approachable but also aligns it with the standard rules of exponent manipulation. Understanding how to handle negative exponents is a fundamental skill in algebra, and this step exemplifies its importance in solving exponential equations. The ability to transform negative exponents into positive ones is a key technique that will be used repeatedly throughout the solution process. Now, with the negative exponent resolved, our equation looks cleaner and more straightforward, paving the way for the next steps in finding the value of x. This initial simplification sets the stage for applying further exponent rules and algebraic manipulations to ultimately solve the equation.
So, the equation now becomes:
(3/4)³ × (3/4)⁷ = (9/16)ˣ
Step 2: Apply the Product of Powers Rule
Now that we've eliminated the negative exponent, the next step involves simplifying the left side of the equation. We have (3/4)³ × (3/4)⁷. Here, we can apply the product of powers rule, which states that when multiplying powers with the same base, you add the exponents: aᵐ × aⁿ = aᵐ⁺ⁿ. In this case, our base is (3/4), and our exponents are 3 and 7. By adding the exponents, we get 3 + 7 = 10. This simplifies the left side of the equation to (3/4)¹⁰. This step is crucial as it consolidates the two exponential terms into a single term, making the equation significantly simpler. The product of powers rule is a cornerstone of exponent manipulation, and its application here demonstrates its power in simplifying complex expressions. By reducing the two terms into one, we've made the equation easier to compare with the right side, which will be essential in the subsequent steps. This simplification not only streamlines the equation but also highlights the elegance and efficiency of exponent rules in mathematical problem-solving. Now, with the left side simplified, we can focus on manipulating the right side of the equation to have a comparable base, bringing us closer to isolating x and finding its value. This step reinforces the importance of recognizing and applying the appropriate exponent rules to effectively solve equations.
Thus, the equation is further simplified to:
(3/4)¹⁰ = (9/16)ˣ
Step 3: Express the Right Side with the Same Base
To solve for x, we need to express both sides of the equation with the same base. Currently, the left side has a base of (3/4), while the right side has a base of (9/16). Notice that 9/16 can be expressed as the square of 3/4, since (3/4)² = (3²/4²) = 9/16. Therefore, we can rewrite the right side of the equation, (9/16)ˣ, as ((3/4)²)ˣ. This transformation is a key step in solving the equation because it allows us to compare the exponents directly. By expressing both sides with the same base, we set the stage for equating the exponents and solving for x. This step demonstrates the importance of recognizing relationships between numbers and their powers. The ability to identify that 9/16 is the square of 3/4 is crucial for simplifying the equation and making it solvable. This skill is invaluable in algebra and other mathematical contexts. Transforming the base on the right side not only simplifies the equation but also showcases the interconnectedness of mathematical concepts. With both sides of the equation now sharing the same base, we're on the verge of isolating x and determining its value. This step highlights the strategic thinking involved in solving exponential equations, where manipulating expressions to achieve a common base is a fundamental technique.
Now, the equation looks like this:
(3/4)¹⁰ = ((3/4)²)ˣ
Step 4: Apply the Power of a Power Rule
In this step, we will simplify the right side of the equation further. We have ((3/4)²)ˣ. According to the power of a power rule, when raising a power to another power, you multiply the exponents: (aᵐ)ⁿ = aᵐⁿ. Applying this rule to our expression, we multiply the exponents 2 and x, resulting in (3/4)²ˣ. This simplification is crucial because it allows us to express the right side of the equation in a form that directly corresponds to the left side. By applying the power of a power rule, we've transformed a complex exponential expression into a simpler one, making it easier to compare with the left side of the equation. This step demonstrates the elegance and efficiency of exponent rules in streamlining mathematical expressions. The power of a power rule is a fundamental concept in algebra, and its application here showcases its importance in simplifying equations. By reducing the complexity of the right side, we've moved closer to isolating x and finding its value. This step reinforces the importance of mastering exponent rules as a key tool in solving mathematical problems. Now, with both sides of the equation expressed with the same base and simplified using exponent rules, we're ready to equate the exponents and solve for the unknown variable. This step underscores the strategic approach involved in solving exponential equations, where simplification and manipulation are essential for reaching the solution.
Thus, the equation now becomes:
(3/4)¹⁰ = (3/4)²ˣ
Step 5: Equate the Exponents and Solve for x
Now that both sides of the equation have the same base, (3/4), we can equate the exponents. This is a fundamental principle in solving exponential equations: if aᵐ = aⁿ, then m = n. In our equation, (3/4)¹⁰ = (3/4)²ˣ, so we can equate the exponents to get 10 = 2x. This step is a critical turning point in the solution process, as it transforms the exponential equation into a simple algebraic equation. By equating the exponents, we've eliminated the exponential terms and focused on a linear equation that is straightforward to solve. This transition highlights the power of algebraic manipulation in simplifying complex problems. To solve for x, we need to isolate it. We can do this by dividing both sides of the equation by 2: 10/2 = 2x/2. This gives us 5 = x. Therefore, the solution to the equation is x = 5. This final step demonstrates the culmination of all the previous steps, where each simplification and transformation led us closer to the solution. The ability to equate exponents and solve for the unknown is a crucial skill in algebra, and this step exemplifies its importance in solving exponential equations. With x now determined, we've successfully navigated the equation and arrived at the solution. This process underscores the importance of understanding exponent rules and algebraic principles in tackling mathematical challenges.
Therefore,
x = 5
Conclusion
In conclusion, we have successfully solved the equation (3/4)³ × (4/3)⁻⁷ = (9/16)ˣ by systematically applying the rules of exponents and algebraic manipulation. We began by understanding the basics of exponents, including the product of powers rule, the quotient of powers rule, the power of a power rule, and the concept of negative exponents. Each step was carefully executed, starting with simplifying the negative exponent, applying the product of powers rule, expressing the right side with the same base, and using the power of a power rule. Finally, we equated the exponents and solved for x, arriving at the solution x = 5. This step-by-step approach not only allowed us to find the solution but also reinforced the importance of understanding and applying fundamental mathematical principles. Solving this equation required a comprehensive understanding of exponent rules and algebraic techniques. The process demonstrated the importance of simplifying expressions, manipulating equations, and applying the correct rules in the appropriate order. Each step built upon the previous one, ultimately leading us to the solution. This exercise highlights the power of a structured approach in tackling mathematical problems. By breaking down the equation into manageable steps, we were able to navigate through the complexities and arrive at a clear and concise answer. The skills and techniques used in this solution are applicable to a wide range of mathematical problems, making this a valuable learning experience. Mastering these concepts will undoubtedly enhance your ability to solve similar equations and tackle more advanced mathematical challenges. This journey through the solution process underscores the beauty and logic of mathematics, where each step is guided by clear rules and principles, ultimately leading to a satisfying conclusion.
