Simplifying The Base Of A Function: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into a fun problem that's all about simplifying the base of a function. Specifically, we're tackling the function f(x)=rac{1}{4}(\sqrt[3]{108})^x and figuring out its simplified base. This might seem a bit tricky at first, but trust me, with a little bit of algebraic know-how, we can crack this one easily. Let's break down the process step by step, making sure everyone's on the same page. Ready? Let's go!

Understanding the Basics: What's a Base Anyway?

Before we get our hands dirty with the math, let's quickly review what a base is. In the function $f(x) = a^x$, 'a' is the base. Essentially, the base is the number that's being raised to the power of x. Understanding the base is crucial because it dictates how quickly the function grows or decays. A base greater than 1 means the function increases as x increases (exponential growth), a base between 0 and 1 means the function decreases as x increases (exponential decay), and a base of 1 means the function stays constant. Knowing this helps us visualize and understand the function's behavior. In our case, the base we're dealing with is not immediately obvious; it's tucked inside a cube root and a power, so our mission is to simplify that and find the 'true' base.

So, why is simplifying the base important? Well, it makes it easier to analyze the function's behavior. A simplified base can help us quickly determine if the function is growing, decaying, or remaining constant. It simplifies calculations, and it gives us a clearer understanding of the function's properties. For instance, knowing the simplified base helps us to easily determine the doubling time (for growth) or the half-life (for decay) of the function. Therefore, simplifying the base is a fundamental skill in algebra and calculus, especially when dealing with exponential functions. It allows for easier comparisons, calculations, and a deeper understanding of the function's properties.

Now, let's move on to the actual simplification of our given function. We'll follow a series of logical steps to get to the answer, making sure each step is clear and easy to follow. Don't worry if it looks complicated initially; we'll take it one step at a time, breaking down the problem into smaller, manageable parts. By the end, you'll have a solid grasp of how to simplify the base of a function like this. Remember, the key is to understand each step and to practice with similar problems to build your confidence and skills.

Decoding the Function: Breaking Down the Components

Alright, let's dissect our function: f(x)=rac{1}{4}(\sqrt[3]{108})^x. The function has a few key parts: the constant rac{1}{4}, the cube root of 108 ($\sqrt[3]{108}$), and the variable 'x' that's the exponent. Our main focus here is the cube root of 108, which we need to simplify. First things first, let's rewrite the function to make it a bit more manageable. We'll start by focusing on the term $(\sqrt[3]{108})^x$. This can be rewritten using the properties of exponents. Recall that $\sqrt[3]{108}$ is the same as $108^{\frac{1}{3}}$. Therefore, we can express $(\sqrt[3]{108})^x$ as $(108^{\frac{1}{3}})^x$. This might seem like a small change, but it sets the stage for the next steps.

Next, we'll use another property of exponents: $(a^b)^c = a^{b \cdot c}$. Applying this to our expression, we get $108^{\frac{1}{3} \cdot x}$, which simplifies to $108^{\frac{x}{3}}$. This form is much easier to work with. Our function now looks like $f(x) = \frac{1}{4} \cdot 108^{\frac{x}{3}}$. This rewriting has helped us in the journey toward a simplified base. But before we get ahead of ourselves, we still need to simplify the base. This simplification is usually done by prime factorization. By doing this, we can easily spot the numbers that are perfect cubes. This is because perfect cubes, when factored, will always have factors that appear in groups of three. Let’s prime factorize 108 now! Remember, prime factorization means breaking down the number into a product of prime numbers (numbers that can only be divided by 1 and themselves).

Let’s start prime factorizing 108. 108 can be divided by 2, giving us 54. Then, 54 can be divided by 2, resulting in 27. 27 can be divided by 3, resulting in 9. 9 can be divided by 3, resulting in 3. And finally, 3 can be divided by 3, resulting in 1. So, the prime factorization of 108 is $2 \cdot 2 \cdot 3 \cdot 3 \cdot 3$, or $2^2 \cdot 3^3$. Using this prime factorization will help us simplify our function. From this factorization, we know the cube root of 108 can also be expressed as $\sqrt[3]{2^2 \cdot 3^3}$. This will allow us to simplify the expression by taking out the cube. Let's get to the next step!

Simplifying the Cube Root and Finding the Base

Okay, we're on the home stretch now! We've got the function $f(x) = \frac{1}{4} \cdot 108^{\frac{x}{3}}$ and we've determined that $108 = 2^2 \cdot 3^3$. Now, substitute the prime factorization back into our function. We have $f(x) = \frac{1}{4} \cdot (2^2 \cdot 3^3)^{\frac{x}{3}}$. Let’s break down the term $(2^2 \cdot 3^3)^{\frac{x}{3}}$ using the power of a product rule, which states that $(ab)^c = a^c \cdot b^c$. Applying this, we get $(2^2)^{\frac{x}{3}} \cdot (3^3)^{\frac{x}{3}}$.

Next, apply the power of a power rule, $(a^b)^c = a^{b \cdot c}$, to each of the terms. For $(2^2)^{\frac{x}{3}}$, this gives us $2^{\frac{2x}{3}}$. For $(3^3)^{\frac{x}{3}}$, this gives us $3^x$. Thus, our function now looks like $f(x) = \frac{1}{4} \cdot 2^{\frac{2x}{3}} \cdot 3^x$. But remember, our goal is to express our function in the form $f(x) = a^x$. We can do this with a few more steps. Notice that the exponent on the 2 is $\frac{2x}{3}$, which we can rewrite as $\frac{2}{3}x$. Then, we can rewrite the entire term as $(2^{\frac{2}{3}})^x$. This implies that we need to change $3^x$ into a form that has the same base as $2^{\frac{2}{3}}$.

Now, let's address the final step. We can't directly combine the terms $2^{\frac{2x}{3}}$ and $3^x$ because they don't have the same base. However, we can focus on simplifying the cube root from the original question. If we go back to our starting point, $(\sqrt[3]{108})^x$, and substitute our prime factorization, we get $(\sqrt[3]{2^2 \cdot 3^3})^x$. Simplifying this, we can take the cube root of $3^3$, which is just 3. Then, the cube root of $2^2$ will remain as $\sqrt[3]{4}$. Now, this means our base is $3\sqrt[3]{4}$. With that, we have found our simplified base. Thus, the simplified form of $f(x)$ becomes $f(x) = \frac{1}{4} \cdot (3\sqrt[3]{4})^x$. The base of the function is, therefore, $3\sqrt[3]{4}$.

Choosing the Right Answer

Now, looking back at our options: A. 3, B. $3\sqrt[3]{4}$, C. $6\sqrt[3]{3}$, and D. 27. After simplifying, we have found that the base is $3\sqrt[3]{4}$. So the correct answer is B. Easy peasy, right? We've successfully simplified the base of our function. The key was to break down the original function into smaller, more manageable pieces using the properties of exponents and prime factorization. We've shown how understanding these tools can help us simplify and manipulate complex-looking functions. By systematically breaking down the problem and applying the rules of algebra, we were able to find the simplified base. This approach is not only useful for this specific problem but can also be applied to many other similar problems. Great job, everyone!