Simplified Base Of F(x) = (1/4)(³√108)^x A Comprehensive Guide

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Introduction

In the realm of mathematics, understanding functions is crucial, and one key aspect is identifying and simplifying the base of exponential functions. Exponential functions, characterized by the form f(x) = a^x, where a is the base, play a significant role in various mathematical models and real-world applications. In this detailed exploration, we aim to dissect the function f(x) = (1/4)(³√108)^x and determine its simplified base. This process involves understanding radicals, exponents, and simplification techniques. By simplifying the base, we gain a clearer understanding of the function's behavior and its properties. This article will guide you through the steps required to simplify the given function, providing a comprehensive explanation of each stage.

Breaking Down the Function

To begin, let's take a closer look at the function at hand: f(x) = (1/4)(³√108)^x. This function has two primary components: the constant coefficient (1/4) and the exponential term (³√108)^x. The exponential term is where our main focus lies, as it contains the base we need to simplify. The cube root of 108 (³√108) is the crucial part of the base that requires simplification. Before we can raise it to the power of x, we need to express it in its simplest form. This involves finding the prime factorization of 108 and extracting any perfect cube factors. Understanding this initial breakdown is essential, as it sets the stage for the subsequent simplification steps. Without this preliminary analysis, it would be challenging to identify the components that need to be simplified. Recognizing the cube root and its importance in determining the base is the first step towards a successful simplification.

Prime Factorization of 108

The next step in simplifying the base involves finding the prime factorization of 108. Prime factorization is the process of breaking down a number into its prime factors, which are the prime numbers that multiply together to give the original number. For 108, we can start by dividing it by the smallest prime number, 2. We get 108 = 2 × 54. Now we continue factoring 54, which is also divisible by 2, resulting in 54 = 2 × 27. Next, we consider 27, which is not divisible by 2 but is divisible by 3. We have 27 = 3 × 9, and finally, 9 = 3 × 3. Putting it all together, the prime factorization of 108 is 2 × 2 × 3 × 3 × 3, or 2² × 3³. This prime factorization is the cornerstone for simplifying the cube root. By expressing 108 in terms of its prime factors, we can easily identify the cubic components that can be extracted from the cube root. This step is critical, as it transforms the initial radical expression into a form that is easier to manipulate and simplify.

Simplifying the Cube Root

With the prime factorization of 108 as 2² × 3³, we can now simplify the cube root. The cube root of 108, denoted as ³√108, can be rewritten as ³√(2² × 3³). When simplifying radicals, we look for factors that can be taken out of the radical. In this case, we have 3³ under the cube root. Since the cube root of 3³ is simply 3, we can extract 3 from the cube root. This leaves us with 3 × ³√2². The term 2² remains under the cube root because it does not have a perfect cube factor. Therefore, ³√2² can be written as ³√4. The simplified form of ³√108 is thus 3 × ³√4. This simplification is a crucial step in determining the base of the exponential function. By extracting the perfect cube factor, we have transformed the original radical expression into a more manageable form. This simplified expression will be used in the subsequent steps to find the simplified base of the function. The process of simplifying cube roots involves identifying and extracting perfect cube factors, which is a fundamental technique in radical simplification.

Rewriting the Function

Now that we have simplified the cube root of 108 to 3 × ³√4, we can rewrite the original function f(x) = (1/4)(³√108)^x. Substituting the simplified cube root, we get f(x) = (1/4)(3 × ³√4)^x. To further simplify the function, we can apply the exponent x to both factors inside the parentheses. This gives us f(x) = (1/4) × 3^x × (³√4)^x. The function now consists of a constant term (1/4), an exponential term with base 3 (3^x), and another exponential term involving a cube root ((³√4)^x). At this stage, it is useful to rewrite the cube root term using fractional exponents. Recall that the nth root of a number can be expressed as a fractional exponent with 1/n as the exponent. Thus, ³√4 can be written as 4^(1/3). Substituting this into our function, we get f(x) = (1/4) × 3^x × (4(1/3))x. This rewriting of the function is a crucial step in simplifying the base, as it allows us to combine exponential terms and further simplify the expression. By converting the radical to a fractional exponent, we can apply exponent rules to combine and simplify the terms more effectively.

Applying Exponent Rules

To continue simplifying the function, we need to apply exponent rules. Specifically, we will focus on the term (4(1/3))x. According to the power of a power rule, which states that (am)n = a^(m × n), we can rewrite (4(1/3))x as 4^((1/3)x). Now our function looks like f(x) = (1/4) × 3^x × 4^((1/3)x). Next, we can express 4 as 2², so 4^((1/3)x) becomes (2²)^((1/3)x). Applying the power of a power rule again, we get 2^((2/3)x). Substituting this back into our function, we have f(x) = (1/4) × 3^x × 2^((2/3)x). At this point, it is essential to recognize that we are trying to find a single simplified base. To do this, we need to combine the exponential terms in a way that allows us to express the function in the form f(x) = a^x. This involves carefully applying exponent rules and algebraic manipulations to achieve a single base. The strategic use of exponent rules is critical in this simplification process, allowing us to combine terms and identify the simplified base.

Combining Exponential Terms

Our function is now in the form f(x) = (1/4) × 3^x × 2^((2/3)x). To find a simplified base, we need to combine the exponential terms. We can rewrite the function as f(x) = (1/4) × (3 × 2(2/3))x. This step involves recognizing that if we have two exponential terms with the same exponent, we can multiply their bases and raise the result to that exponent. In this case, we have 3^x and 2^((2/3)x), so we combine their bases by multiplying 3 and 2^(2/3). This gives us a new base of 3 × 2^(2/3). Now the function is f(x) = (1/4) × (3 × 2(2/3))x. To completely simplify the function, we need to address the constant coefficient (1/4). We can rewrite this as f(x) = (1/4) × (3 × ∛4)^x. However, the question asks for the simplified base, which is the term raised to the power of x. Therefore, the simplified base is 3 × 2^(2/3) or 3 × ∛4. This combination of exponential terms is a critical step in the simplification process. By recognizing the common exponent x, we can combine the bases and express the function in a form that clearly shows the simplified base. The ability to manipulate exponential terms and combine them effectively is a key skill in simplifying exponential functions.

The Simplified Base

After carefully applying exponent rules and combining terms, we have arrived at the simplified base of the function. From our previous steps, the function f(x) = (1/4)(³√108)^x has been rewritten as f(x) = (1/4) × (3 × 2(2/3))x. The base of the exponential term is 3 × 2^(2/3). This can also be expressed as 3 × ∛4, where ∛4 represents the cube root of 4. Thus, the simplified base of the function is 3 × 2^(2/3) or 3∛4. This final answer is the culmination of all the simplification steps we have taken, including prime factorization, simplifying radicals, applying exponent rules, and combining exponential terms. Identifying the simplified base is crucial for understanding the behavior of the function and its applications in various mathematical contexts. The simplified base allows us to analyze the function's growth rate, transformations, and other properties more effectively. This process of simplification not only provides a concise form of the base but also enhances our comprehension of the function's characteristics and its role in mathematical models.

Conclusion

In conclusion, the process of finding the simplified base of the function f(x) = (1/4)(³√108)^x involves several key steps. We began by breaking down the function and identifying the need to simplify the cube root of 108. We then performed prime factorization to express 108 as 2² × 3³, which allowed us to simplify the cube root to 3 × ³√4. Next, we rewrote the function using this simplified cube root and applied exponent rules to combine exponential terms. Through these steps, we identified the simplified base as 3 × 2^(2/3) or 3∛4. This exercise demonstrates the importance of understanding exponents, radicals, and simplification techniques in mathematics. Simplifying the base of a function not only makes it easier to work with but also provides deeper insights into its properties and behavior. The skills and techniques used in this simplification process are applicable to a wide range of mathematical problems and are essential for anyone studying functions and their applications. By mastering these concepts, one can approach more complex mathematical challenges with confidence and precision.