Find Exponential Function With Base 4√[3]4: A Step-by-Step Guide

Jul 11, 2025 by ADMIN 65 views

Decoding the Exponential Function with Base 4√[3]4

In the realm of exponential functions, understanding the base is crucial for analyzing their behavior and properties. When presented with several exponential functions, each with a seemingly complex base, identifying the function that simplifies to a specific base requires a careful application of exponent rules and simplification techniques. This article will delve into the process of simplifying exponential functions and determining which one has a base of 4√[3]4. This exploration will not only help in solving the given problem but also enhance your understanding of exponential functions in general.

Understanding Exponential Functions and Base Simplification

Before we dive into the specific functions, let's recap the fundamental concepts of exponential functions and base simplification. An exponential function is generally represented as f(x) = a^x, where 'a' is the base and 'x' is the exponent. The base 'a' determines the rate at which the function grows or decays. Simplifying the base often makes it easier to compare and analyze different exponential functions.

When dealing with bases involving radicals and exponents, the following rules are essential:

(a^m)n = a^(m*n): This rule states that when raising a power to another power, you multiply the exponents.
a^(m/n) = √n: This rule connects fractional exponents with radicals, where 'm' is the power and 'n' is the root.
a^m * a^n = a^(m+n): This rule applies when multiplying exponential terms with the same base; you add the exponents.

By applying these rules judiciously, we can simplify complex bases and identify equivalent exponential functions. In the context of our problem, we aim to transform the given bases into the form 4√[3]4, which can also be expressed as 4 * 4^(1/3) or 4^(4/3). This is our target form for simplification.

Analyzing the Given Functions

Let's now examine the given functions one by one, simplifying their bases and comparing them to our target base of 4√[3]4:

Function 1: f(x) = 2(√[3]16)^x

To simplify the base of this function, we start by expressing 16 as a power of 2, since 16 = 2^4. Therefore, the base becomes √3. We can rewrite this using fractional exponents as (2⁴⁾(1/3), which simplifies to 2^(4/3). Now, we have the base as 2^(4/3). Our goal is to express this in the form of 4√[3]4 or 4^(4/3). Let's rewrite 4^(4/3) as (2²⁾(4/3) which equals to 2^(8/3). Comparing 2^(4/3) with 2^(8/3), we see they are not the same. Alternatively, we can rewrite 2^(4/3) as 2 * 2^(1/3). This is not in the form of 4√[3]4, so the base of this function does not simplify to our target.

Function 2: f(x) = 2(√[3]64)^x

For the second function, the base is √[3]64. We know that 64 is a perfect cube, specifically 64 = 4^3. Thus, the base simplifies to √3, which is simply 4. The function becomes f(x) = 2 * 4^x. This is not in the desired form of 4√[3]4, which is 4 * 4^(1/3) or 4^(4/3). Therefore, this function does not have the target base.

Function 3: f(x) = 4(√[3]16)^(2x)

This function has a slightly more complex exponent, 2x, but the process remains the same. The base is √[3]16, which we already simplified in the first function as 2^(4/3). The function can be rewritten as f(x) = 4 * (2^(4/3))(2x). Focusing on the base, we have to simplify (2^(4/3))2. Applying the power of a power rule, we get 2^((4/3)*2) = 2^(8/3). Now, we have the function as f(x) = 4 * (2^(8/3))x. To see if this matches our target base, we need to express 2^(8/3) in terms of 4. We know that 4^(4/3) is 4√[3]4. So, let's try to express 2^(8/3) in terms of 4^(4/3). We know 4=2^2, thus 4^(4/3) = (2²⁾(4/3) = 2^(8/3). This indicates that the base (√[3]16)^2 = 2^(8/3) is equal to 4^(4/3) = 4√[3]4. Therefore, the simplified function is f(x) = 4 * (4^(4/3))x = 4 * (4√[3]4)^x. If we further simplify we get 4^(1) * 4^(4x/3). Thus, this function's base 4√[3]4 matches our target.

Function 4: f(x) = 4(√[3]64)^(2x)

Lastly, we examine the fourth function. The base is √[3]64, which we simplified in the second function as 4. The function becomes f(x) = 4 * (4)^(2x). Simplifying the base with the exponent, we have 4^2 = 16. If we simplify it to f(x) = 4 * 16^x, this is clearly not in the form of 4√[3]4. Hence, this function does not have the target base.

Conclusion: Identifying the Function with Base 4√[3]4

Through a systematic simplification process, we have analyzed each of the given functions. By applying exponent rules and expressing the bases in a simplified form, we were able to identify the function that has a base of 4√[3]4. Function 3, f(x) = 4(√[3]16)^(2x), simplified to 4 * (4√[3]4)^x, thus confirming that its base matches our target. This exercise demonstrates the importance of understanding exponent rules and simplification techniques when working with exponential functions.

Which of the following functions has a simplified base equal to 4 multiplied by the cube root of 4 (4√[3]4)?

Find Exponential Function with Base 4√[3]4 A Step-by-Step Guide