Finding The Rate Of Increase For F(x) = (1/3)(24^(1/3))^(2x)

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Hey guys! Let's dive into a cool math problem today. We're going to figure out the rate of increase for the function f(x) = (1/3)(24(1/3))(2x). This might look a little intimidating at first, but don't worry, we'll break it down step by step so it's super easy to understand. We'll explore exponential functions, rates of change, and how to simplify expressions with radicals and exponents. By the end of this article, you'll not only know the answer but also understand the process behind it. So, grab your calculators (or not, if you're feeling like a math whiz!), and let's get started!

Understanding the Function

Before we jump into calculating the rate of increase, let's take a closer look at the function f(x) = (1/3)(24(1/3))(2x). This is an exponential function, which means the variable x is in the exponent. Exponential functions have a general form of f(x) = a * b^(cx), where:

  • a is the initial value or the coefficient.
  • b is the base, which determines the rate of growth or decay.
  • c is a constant that affects the rate.

In our case, a = 1/3, and the base looks a bit complex, but it's (24(1/3))(2). We'll simplify this in a bit. Recognizing this form is the first key step. Why is it important to recognize the form? Because the base b is directly related to the rate of increase. If b is greater than 1, the function is increasing; if b is between 0 and 1, it's decreasing. Understanding this foundational concept makes the rest of the problem much clearer. Exponential functions are used everywhere, from calculating compound interest to modeling population growth, so getting a handle on them is a huge win.

Simplifying the Base

The heart of this problem lies in simplifying the base of the exponential function. Remember, our base is (24(1/3))(2). To make this easier to work with, we'll use a couple of exponent rules:

  1. (a(m))n = a^(mn)* (Power of a Power)
  2. a^(m) * a^(n) = a^(m+n) (Product of Powers - although we won't directly use it here, it's good to remember!)

Applying the first rule, we can rewrite (24(1/3))(2) as *24^((1/3)2), which simplifies to 24^(2/3). Now, we need to simplify 24^(2/3) further. The exponent 2/3 can be thought of as taking the cube root (the denominator 3) and then squaring the result (the numerator 2). So, we have (24(1/3))2. Let's think about the cube root of 24. We can break 24 down into its prime factors: 24 = 2 * 2 * 2 * 3 = 2^3 * 3. Therefore, 24^(1/3) = (2^3 * 3)^(1/3). Using another exponent rule, (a * b)^n = a^n * b^n, we get (23)(1/3) * 3^(1/3) = 2 * 3^(1/3). Now, we square this result: (2 * 3(1/3))2 = 2^2 * (3(1/3))2 = 4 * 3^(2/3). So, the simplified base is 4 * 3^(2/3). See how breaking it down step-by-step makes it less scary? We've gone from a complex-looking expression to something much more manageable. This skill of simplifying expressions is absolutely crucial in math, and it's something you'll use again and again.

Determining the Rate of Increase

Okay, we've simplified the base to 4 * 3^(2/3). Remember, the base of the exponential function is what determines the rate of increase. So, b = 4 * 3^(2/3). Now, we need to figure out what that actually means in terms of a rate. We have 3^(2/3), which is the cube root of 3 squared. Let's think about this in simpler terms. 3^(2/3) can also be expressed as (∛3)^2. Since we are looking for the rate of increase, we need to focus on the value of the base, which we’ve now simplified to 4 * 3^(2/3) or 4 * (∛3)^2. This form helps us see the components more clearly. The coefficient 4 is a straightforward multiplier, and the term (∛3)^2 represents the core exponential growth factor. We can rewrite 3^(2/3) as the cube root of 9 (∛9). So, our base is 4 * ∛9. Therefore, the rate of increase is 4 * ∛9. This matches one of the options given, but it's super important to understand why this is the rate of increase. The base of the exponential function directly tells us how much the function multiplies its value for every unit increase in x. In this case, for every increase of 1 in x, the function's value is multiplied by 4 * ∛9. This is the essence of exponential growth.

The Final Answer and Why It Matters

So, after all that simplification and calculation, we've found that the rate of increase for the function f(x) = (1/3)(24(1/3))(2x) is 4 * ∛9, which corresponds to option D. But the real win here isn't just getting the right answer; it's understanding the process. We broke down a complex-looking function, simplified exponents and radicals, and connected the base of an exponential function to its rate of increase. These are skills that will serve you well in all sorts of math problems (and even in real-world situations!). Understanding exponential growth is crucial in many fields, from finance to biology. Knowing how to manipulate exponents and radicals is a foundational math skill. So, pat yourselves on the back, guys! You've tackled a challenging problem and learned some valuable concepts along the way. Keep practicing, keep exploring, and remember, math can be fun!