Equivalent Functions: Solving The Fourth Root Equation
Hey math enthusiasts! Let's dive into a fun problem involving exponents and radicals. We're tasked with finding which functions are equivalent to f(x) = √[4]{162}ˣ. This might seem a bit tricky at first, but trust me, we'll break it down step-by-step. The key here is to understand the properties of exponents and how they interact with radicals. We'll rewrite the original function and then compare it with the given options to see which ones match. So, grab your calculators (or your brains!) and let's get started. By the end, you'll be a pro at identifying equivalent exponential functions. Ready? Let's go!
Understanding the Core Equation: f(x) = √[4]{162}ˣ
Alright, guys, let's start by unpacking the original equation, f(x) = √[4]{162}ˣ. The expression √[4]{162} is the fourth root of 162. Remember that a fourth root means finding a number that, when raised to the power of 4, equals 162. We can rewrite this using fractional exponents. The fourth root of a number is the same as raising that number to the power of 1/4. So, √[4]{162} is the same as 162^(1/4). Therefore, our original function can be rewritten as f(x) = (162^(1/4))ˣ. Now, using the power of a power rule in exponents, which states that (am)n = a^(m*n), we can simplify this further. This rule tells us that when you have a power raised to another power, you multiply the exponents. In our case, we have (162^(1/4))ˣ, so we can rewrite this as 162^(x/4). This form is crucial because it helps us directly compare our original function with the answer choices. This is the bedrock of our problem, and understanding this transformation is critical to correctly answering the question. The main takeaway here is the interplay between radicals and exponents – they are two sides of the same coin and can be converted from one form to the other to simplify calculations and comparisons. Got it? Awesome, let's look at the options.
Breaking Down Option A: f(x) = 162^(x/4)
Let's analyze the first option: f(x) = 162^(x/4). As we discussed, our original function f(x) = √[4]{162}ˣ can be simplified to 162^(x/4). This option directly matches the simplified form of our original function. Since the base and the exponent's relationship are identical to our simplified equation, this option is, without a doubt, equivalent. No further calculations are needed here; the functions are literally identical. So, we've found our first correct answer! Keep in mind that understanding how to rewrite radical expressions with fractional exponents is the foundation of correctly solving this kind of problem. This is a clear example of applying the rules of exponents to simplify and find equivalents. The power of rewriting expressions is that it helps to transform them into more familiar and manageable forms, making comparison and identification of equivalence far easier. Therefore, option A is confirmed to be equivalent to the original function.
Examining Option B: f(x) = (3√[4]{2})ˣ
Now, let's move on to option B: f(x) = (3√[4]{2})ˣ. This one looks a bit different, so let's try to rewrite it and see if we can make it match our original simplified form. First, let's look at the base of the exponential function, which is 3√[4]{2}. We can rewrite this using the properties of radicals and exponents. √[4]{2} is the same as 2^(1/4). So, we have 3 * 2^(1/4). To compare this effectively, we want to see if we can rewrite it to a base related to 162. Remember, our initial equation, 162^(x/4), is what we are trying to match. Since 162 can be factored into prime numbers (162 = 2 * 3⁴), we can rewrite the base of our original equation as (2 * 3⁴)^(1/4) = 2^(1/4) * 3^(4/4) = 2^(1/4) * 3. So the key here is to realize that 3√[4]{2} is essentially the fourth root of 162. So we can say that (3 * 2^(1/4)) = (3 * √[4]{2}). The base of option B is 3√[4]{2}. If we raise this to the power of x, we get (3√[4]{2})ˣ. To determine if this is equivalent, we can convert the 3√[4]{2} part to an exponential form: 3 * 2^(1/4). If we raise this to the power of x, the function becomes (3 * 2^(1/4))ˣ = 3ˣ * 2^(x/4). This is not the same as 162^(x/4), because it's not the same base with respect to the exponent x. Therefore, Option B is incorrect because its base cannot be converted to the same base as option A.
Deconstructing Option C: f(x) = 9√[4]{2}ˣ
Let's give option C a shot: f(x) = 9√[4]{2}ˣ. We want to determine if this is equivalent to our simplified original function, which we know is 162^(x/4). Again, let's focus on the base. We have 9√[4]{2}. We can rewrite 9 as 3². So, our base becomes 3² * 2^(1/4). When we raise this to the power of x, the function becomes (3² * 2^(1/4))ˣ = 3^(2x) * 2^(x/4). This looks very different from 162^(x/4). As before, we are looking for a function that simplifies to the same base and exponent as our initial equation (162^(x/4)). Since option C contains the term 3^(2x), it cannot be equivalent to our original function. Remember the laws of exponents and how different bases affect the resulting expression. Thus, option C is incorrect because the base cannot be converted to the same base as option A.
Scrutinizing Option D: f(x) = 162^(4/x)
Alright, let's tackle option D: f(x) = 162^(4/x). This looks interesting because it has 162 as the base, which is good. However, the exponent is where the trouble lies. Our simplified original function has an exponent of x/4, and this option has an exponent of 4/x. For these to be equal, it must be true that x/4 = 4/x. However, for most values of x, these expressions are not equivalent. In fact, if we solve for x, we would get x² = 16, or x = ±4. However, we're looking for an equivalence for all values of x. Because the exponent is different, option D is incorrect. Therefore, option D cannot be equivalent to the original function.
Analyzing Option E: f(x) = [3(2^(1/4))]ˣ
Finally, let's analyze option E: f(x) = [3(2^(1/4))]ˣ. This one looks promising! Remember that we simplified our original function to 162^(x/4) = (162^(1/4))ˣ. Let's see if we can relate option E to this. We can rewrite 162 as 2 * 3⁴, thus 162^(1/4) = (2 * 3⁴)^(1/4) = 2^(1/4) * 3^(4/4) = 3 * 2^(1/4). Option E is then [3 * 2^(1/4)]ˣ. As we have already shown, 3 * 2^(1/4) = √[4]{162}. Hence, the base of option E is precisely the same as that of our original expression. If we rewrite it in the form of a radical, we get the same thing. This is exactly the same as our original expression. Option E is equivalent to (√[4]{162})ˣ = 162^(x/4). This is the correct answer. The key takeaway here is to simplify and rewrite expressions into their component parts so that you can compare them with your simplified original equation.
Conclusion: Identifying Equivalent Functions
So, guys, after careful analysis, we've identified the equivalent functions. Option A and E are equivalent to the original function, f(x) = √[4]{162}ˣ. This was a great exercise in applying the properties of exponents and radicals! Remember, when dealing with these types of problems, the most important thing is to rewrite the expressions in a way that allows for easy comparison. Keep practicing, and you'll become a pro at identifying equivalent exponential functions. Good job everyone!