Correctly Simplified Exponential Functions: Check Your Answers

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Hey guys! Let's dive into the world of exponential functions and see how to simplify them correctly. It's super important to understand these simplifications, especially when you're dealing with more complex math problems. This article will break down some examples and help you figure out which ones are simplified right. So, grab your thinking caps, and let's get started!

Understanding Exponential Functions

Before we jump into the specific examples, let's quickly recap what exponential functions are all about. An exponential function has the general form f(x) = ab^x*, where a is the initial value, b is the base (a positive number not equal to 1), and x is the exponent. The key here is that the variable x is in the exponent, which means the function grows (or decays) at an exponential rate. When simplifying these functions, we're essentially trying to rewrite them in a more manageable or recognizable form, often by manipulating the base or using exponent rules.

Key Concepts in Exponential Function Simplification

When simplifying exponential functions, there are a few key concepts we need to keep in mind. First, remember the basic exponent rules, such as the power of a power rule ((am)n = a^(mn)*) and the product of powers rule (a^m * a^n = a^(m+n)). These rules are essential for manipulating exponents and simplifying expressions. Second, it's crucial to understand how to rewrite radicals as fractional exponents. For example, the cube root of x can be written as x^(1/3), and the square root of x is x^(1/2). This conversion is often necessary to combine terms or apply exponent rules effectively. Finally, recognizing perfect squares, cubes, and other powers can significantly speed up the simplification process. For instance, knowing that 8 is 2^3 and 16 is 2^4 allows you to quickly rewrite these numbers in exponential form.

Why Simplification Matters

You might be wondering, why bother simplifying exponential functions at all? Well, simplification makes these functions easier to work with in several ways. A simplified function is often easier to graph, as it may be in a standard form that you immediately recognize. Simplification can also make it easier to compare different exponential functions or to identify key features, such as the growth rate or initial value. Moreover, when solving equations involving exponential functions, a simplified form can make the equation much easier to solve. In practical applications, such as modeling population growth or radioactive decay, simplified functions can provide clearer insights into the underlying process. So, mastering the art of simplification is a valuable skill in mathematics and beyond.

Analyzing the Given Exponential Functions

Alright, let's jump into the examples you provided and see which ones have been simplified correctly. We'll go through each one step-by-step, so you can follow along and understand the reasoning behind each simplification. Remember, the goal is to rewrite the function in its simplest form while maintaining its original value.

Example 1: f(x) = 5 √[3]{16}^x = 5(2 √[3]{2})^x

Let's tackle the first function: f(x) = 5 √[3]{16}^x = 5(2 √[3]{2})^x. The key here is to simplify the cube root of 16. We know that 16 can be written as 2^4. So, we have √[3]{16} = √[3]{2^4}. Now, we can rewrite this as 2^(4/3). Breaking this down further, 2^(4/3) = 2^(1 + 1/3) = 2^1 * 2^(1/3) = 2 √[3]{2}. Therefore, the original function f(x) = 5 √[3]{16}^x can indeed be simplified to 5(2 √[3]{2})^x. This simplification is correct.

  • Step-by-step breakdown:
    • Rewrite 16 as 2^4: √[3]{16} = √[3]{2^4}
    • Convert the cube root to a fractional exponent: √[3]{2^4} = 2^(4/3)
    • Separate the exponent: 2^(4/3) = 2^(1 + 1/3)
    • Rewrite as a product: 2^(1 + 1/3) = 2^1 * 2^(1/3)
    • Simplify: 2^1 * 2^(1/3) = 2 √[3]{2}
    • Final simplified form: f(x) = 5(2 √[3]{2})^x

Example 2: f(x) = 2.3(8)^(1/2 x) = 2.3(4)^x

Now, let's look at the second function: f(x) = 2.3(8)^(1/2 x) = 2.3(4)^x. In this case, we need to focus on simplifying the base, which is 8 raised to the power of (1/2)x. We can rewrite 8 as 2^3, so we have (8)^(1/2 x) = (23)(1/2 x). Using the power of a power rule, we get 2^(3 * 1/2 x) = 2^(3/2 x). To get to the simplified form, we need to see if 2^(3/2 x) is equivalent to 4^x. Since 4 is 2^2, we can rewrite 4^x as (22)x = 2^(2x). Comparing 2^(3/2 x) and 2^(2x), we see that they are not the same. Therefore, the simplification f(x) = 2.3(8)^(1/2 x) = 2.3(4)^x is incorrect.

  • Step-by-step breakdown:
    • Rewrite 8 as 2^3: (8)^(1/2 x) = (23)(1/2 x)
    • Apply the power of a power rule: (23)(1/2 x) = 2^(3/2 x)
    • Rewrite 4 as 2^2: 4^x = (22)x = 2^(2x)
    • Compare exponents: 2^(3/2 x) β‰  2^(2x)
    • Conclusion: The simplification is incorrect.

Example 3: f(x) = 81^(x/4) = 3^x

Let's move on to the third function: f(x) = 81^(x/4) = 3^x. Here, we need to simplify 81 raised to the power of x/4. We know that 81 is 3^4, so we can rewrite the function as (34)(x/4). Applying the power of a power rule, we get 3^(4 * x/4) = 3^x. This simplification is correct.

  • Step-by-step breakdown:
    • Rewrite 81 as 3^4: 81^(x/4) = (34)(x/4)
    • Apply the power of a power rule: (34)(x/4) = 3^(4 * x/4)
    • Simplify the exponent: 3^(4 * x/4) = 3^x
    • Conclusion: The simplification is correct.

Example 4: f(x) = 3/4 √{27}^x = 3/4(3 √{3})^x

Finally, let's examine the fourth function: f(x) = 3/4 √{27}^x = 3/4(3 √(3))^x. We need to simplify the square root of 27. We know that 27 is 3^3, so we have √(27) = √(3^3). We can rewrite this as 3^(3/2). Breaking this down further, 3^(3/2) = 3^(1 + 1/2) = 3^1 * 3^(1/2) = 3 √(3). Therefore, the original function f(x) = 3/4 √{27}^x can indeed be simplified to 3/4(3 √(3))^x. This simplification is correct.

  • Step-by-step breakdown:
    • Rewrite 27 as 3^3: √(27) = √(3^3)
    • Convert the square root to a fractional exponent: √(3^3) = 3^(3/2)
    • Separate the exponent: 3^(3/2) = 3^(1 + 1/2)
    • Rewrite as a product: 3^(1 + 1/2) = 3^1 * 3^(1/2)
    • Simplify: 3^1 * 3^(1/2) = 3 √(3)
    • Final simplified form: f(x) = 3/4(3 √(3))^x

Conclusion: Which Functions Were Correctly Simplified?

So, guys, after analyzing each exponential function, we found that the following simplifications were done correctly:

  • f(x) = 5 √[3]{16}^x = 5(2 √[3]{2})^x
  • f(x) = 81^(x/4) = 3^x
  • f(x) = 3/4 √{27}^x = 3/4(3 √(3))^x

The simplification f(x) = 2.3(8)^(1/2 x) = 2.3(4)^x was incorrect. Remember, the key to simplifying exponential functions is to use the exponent rules and rewrite the base in its simplest form. Keep practicing, and you'll become a pro at simplifying these functions in no time!

Final Thoughts and Tips

Simplifying exponential functions might seem tricky at first, but with a solid grasp of exponent rules and a bit of practice, you'll be able to tackle even the most complex expressions. Always remember to break down the problem into smaller steps, focus on simplifying the base, and apply the power of a power rule when necessary. And hey, don't be afraid to double-check your work – math is all about precision! Keep up the great work, and you'll master exponential functions in no time. You got this!