Evaluating G(4/3) For G(x) = (1/64)^x A Comprehensive Guide
In the realm of mathematics, exponential functions play a crucial role in modeling various real-world phenomena, from population growth to radioactive decay. Understanding how to evaluate these functions for specific inputs is essential for solving a wide range of problems. In this article, we will delve into the process of evaluating the exponential function g(x) = (1/64)^x for the input x = 4/3. We will break down the steps involved, providing a clear and concise explanation that will empower you to tackle similar problems with confidence. This article serves as a comprehensive guide, providing not only the solution but also the underlying concepts and techniques necessary for mastering exponential function evaluation.
Understanding Exponential Functions
Before we dive into the specific problem at hand, let's take a moment to solidify our understanding of exponential functions in general. An exponential function is a mathematical function of the form f(x) = a^x, where a is a constant called the base and x is the exponent. The base a must be a positive real number not equal to 1. The exponent x can be any real number. The behavior of an exponential function is heavily influenced by the value of the base a. If a is greater than 1, the function represents exponential growth, meaning that the function's value increases rapidly as x increases. Conversely, if a is between 0 and 1, the function represents exponential decay, meaning that the function's value decreases rapidly as x increases. Exponential functions are used extensively in various fields, including finance, biology, physics, and computer science. For instance, they can model compound interest, population growth, radioactive decay, and the spread of diseases. Their ability to capture rapid growth or decay makes them indispensable tools for analyzing and predicting a wide range of phenomena. Understanding the properties and behavior of exponential functions is therefore crucial for anyone working in these fields.
Problem Statement
Our specific problem involves the exponential function g(x) = (1/64)^x. We are tasked with finding the value of this function when x = 4/3, which means we need to evaluate g(4/3). This problem provides an excellent opportunity to apply our understanding of exponential functions and their properties. We will need to utilize the rules of exponents, particularly those related to fractional exponents and negative bases. By carefully applying these rules, we can simplify the expression and arrive at the final answer. The process of evaluating g(4/3) will not only give us a numerical result but also enhance our understanding of how exponential functions behave with fractional exponents. This type of problem is a common occurrence in algebra and calculus, so mastering the techniques involved is essential for further mathematical studies. The ability to confidently evaluate exponential functions for various inputs is a fundamental skill that will serve you well in many mathematical contexts.
Step-by-Step Solution
To find g(4/3), we substitute x = 4/3 into the function g(x) = (1/64)^x: g(4/3) = (1/64)^(4/3). Now, we need to simplify this expression. The key to simplifying this lies in recognizing that 64 is a perfect cube. Specifically, 64 is equal to 4 cubed (4^3). We can rewrite the expression as g(4/3) = (1/43)(4/3). Next, we can use the property of exponents that states (am)n = a^(m*n). Applying this property, we get g(4/3) = (4(-3))(4/3). Multiplying the exponents, we have g(4/3) = 4^(-3 * 4/3). The 3 in the numerator and denominator cancels out, leaving us with g(4/3) = 4^(-4). Finally, we use the property that a^(-n) = 1/a^n to rewrite the expression as g(4/3) = 1/4^4. Calculating 4 to the power of 4, we get 4^4 = 256. Therefore, g(4/3) = 1/256. This step-by-step breakdown demonstrates how we can systematically simplify an exponential expression with a fractional exponent. Each step utilizes a fundamental property of exponents, allowing us to transform the expression into a more manageable form. The final result, 1/256, is the value of the function g(x) when x = 4/3.
Detailed Explanation of Steps
Let's break down each step in more detail to ensure a complete understanding. First, we substituted x = 4/3 into the function g(x) = (1/64)^x, resulting in g(4/3) = (1/64)^(4/3). This is a straightforward application of function evaluation, where we replace the variable x with the given value. The next crucial step is recognizing that 64 can be expressed as 4 cubed (4^3). This allows us to rewrite the expression as g(4/3) = (1/43)(4/3). This step is important because it sets the stage for simplifying the expression using exponent rules. We then used the property of exponents that states (am)n = a^(m*n). This property allows us to multiply the exponents when we have a power raised to another power. Applying this property, we get g(4/3) = (4(-3))(4/3). Here, we've also used the property that 1/a^n = a^(-n) to rewrite 1/4^3 as 4^(-3). This step is crucial for effectively applying the power of a power rule. Multiplying the exponents, we have g(4/3) = 4^(-3 * 4/3). The 3 in the numerator and denominator cancels out, leaving us with g(4/3) = 4^(-4). This simplification makes the expression much easier to evaluate. Finally, we used the property that a^(-n) = 1/a^n to rewrite the expression as g(4/3) = 1/4^4. This property allows us to deal with negative exponents by taking the reciprocal of the base raised to the positive exponent. Calculating 4 to the power of 4, we get 4^4 = 256. Therefore, g(4/3) = 1/256. This final calculation gives us the numerical value of the function at the given input. By understanding the reasoning behind each step and the properties of exponents used, you can confidently tackle similar problems involving exponential functions.
Alternative Approaches
While the step-by-step solution outlined above is a standard and efficient approach, there are alternative ways to solve this problem that can provide additional insights. One alternative approach involves directly dealing with the fractional exponent. Recall that a fractional exponent can be interpreted as a combination of a root and a power. Specifically, a^(m/n) can be understood as the nth root of a raised to the mth power, or (n√a)^m. Applying this to our problem, (1/64)^(4/3) can be interpreted as the cube root of 1/64, raised to the fourth power. The cube root of 1/64 is 1/4, since (1/4)^3 = 1/64. Therefore, we have (1/4)^4, which is equal to 1/256. This approach directly uses the definition of fractional exponents, providing a different perspective on the problem. Another alternative approach involves expressing 1/64 as a power of 2. We know that 64 is 2^6, so 1/64 is 2^(-6). Therefore, we can rewrite the expression as (2(-6))(4/3). Applying the power of a power rule, we get 2^(-6 * 4/3). Multiplying the exponents, we have 2^(-8). Finally, using the property of negative exponents, we get 1/2^8, which is equal to 1/256. This approach highlights the importance of recognizing common powers and using them to simplify expressions. These alternative approaches not only lead to the same correct answer but also demonstrate the flexibility and interconnectedness of mathematical concepts. By exploring different solution paths, you can deepen your understanding and develop problem-solving skills.
Common Mistakes to Avoid
When evaluating exponential functions, there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure accurate solutions. One common mistake is misinterpreting fractional exponents. Remember that a^(m/n) represents the nth root of a raised to the mth power, not the other way around. For example, 8^(2/3) is the cube root of 8 squared, which is 4, not the square root of 8 cubed. Another common mistake is incorrectly applying the power of a power rule. Ensure that you multiply the exponents correctly when you have a power raised to another power. For example, (am)n = a^(m*n), not a^(m+n). A third mistake involves mishandling negative exponents. Remember that a^(-n) = 1/a^n, which means a negative exponent indicates a reciprocal. For example, 2^(-3) = 1/2^3 = 1/8, not -2^3. Finally, be careful with the order of operations. When simplifying expressions, follow the PEMDAS/BODMAS rule: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This will help you avoid errors in your calculations. By understanding these common mistakes and taking care to avoid them, you can significantly improve your accuracy and confidence in evaluating exponential functions. Practice and careful attention to detail are key to mastering this skill.
Real-World Applications
Exponential functions are not just abstract mathematical concepts; they have numerous applications in the real world. Understanding how to evaluate them is crucial for solving problems in various fields. One prominent application is in finance, where exponential functions are used to model compound interest. The formula for compound interest is A = P(1 + r/n)^(nt), where A is the final amount, P is the principal amount, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the number of years. Evaluating this function for different values of t allows us to project the growth of an investment over time. Another important application is in biology, where exponential functions are used to model population growth. The population growth model is often expressed as P(t) = P0 * e^(kt), where P(t) is the population at time t, P0 is the initial population, e is the base of the natural logarithm (approximately 2.718), and k is the growth rate constant. Evaluating this function for different values of t helps us predict how a population will change over time. Exponential functions also play a crucial role in physics, particularly in the study of radioactive decay. The decay of a radioactive substance follows an exponential decay model, given by N(t) = N0 * e^(-λt), where N(t) is the amount of substance remaining at time t, N0 is the initial amount, e is the base of the natural logarithm, and λ is the decay constant. Evaluating this function allows us to determine the half-life of a radioactive substance, which is the time it takes for half of the substance to decay. These are just a few examples of the many real-world applications of exponential functions. By mastering the skills needed to evaluate them, you can gain a deeper understanding of these applications and make informed decisions in various contexts.
Conclusion
In conclusion, evaluating exponential functions is a fundamental skill in mathematics with wide-ranging applications. In this article, we have explored the process of evaluating the exponential function g(x) = (1/64)^x for the input x = 4/3. We have provided a step-by-step solution, a detailed explanation of each step, and alternative approaches to solving the problem. We have also discussed common mistakes to avoid and highlighted real-world applications of exponential functions. The key to successfully evaluating exponential functions lies in understanding the properties of exponents and applying them systematically. By recognizing perfect squares, perfect cubes, and other common powers, you can simplify expressions and make calculations easier. Remember to pay close attention to the order of operations and to handle negative and fractional exponents correctly. Practice is essential for mastering this skill. By working through various examples and applying the techniques discussed in this article, you can build your confidence and accuracy in evaluating exponential functions. This skill will not only serve you well in your mathematical studies but also in various real-world applications where exponential functions play a crucial role. We hope this comprehensive guide has provided you with the knowledge and tools you need to confidently tackle exponential function evaluation problems.
