When delving into the world of mathematics, exponential functions hold a significant place. They describe scenarios where quantities grow or decay at a rate proportional to their current value. In this comprehensive guide, we will explore the correct simplification of various exponential functions, providing a detailed analysis of each option. Our main goal is to equip you with the skills to confidently tackle these problems. We'll dissect each function, revealing the underlying mathematical principles and showing how to arrive at the simplified forms. Understanding exponential functions is crucial not only in mathematics but also in various real-world applications such as finance, biology, and physics.
A) f(x) = 5 ∛16^x = 5 (2 ∛2)^x
Let's begin by examining the first exponential function: f(x) = 5 ∛16^x. The key to simplifying this function lies in understanding how to express the cube root and the number 16 in terms of their prime factors. We can rewrite 16 as 2^4. Therefore, the function becomes:
f(x) = 5 ∛(24)x
Now, we need to apply the properties of exponents. Specifically, (am)n = a^(m*n). Using this property, we get:
f(x) = 5 ∛2^(4x)
The cube root can be expressed as an exponent of 1/3. Thus, the function can be further rewritten as:
f(x) = 5 (2(4x))(1/3)
Again, applying the property of exponents (am)n = a^(m*n), we multiply the exponents:
f(x) = 5 * 2^(4x/3)
To simplify this further, we can express 2^(4x/3) as 2^(x * 4/3), which is equivalent to 2^(x * (3/3 + 1/3)). Splitting the exponent, we get:
f(x) = 5 * 2^(x * (1 + 1/3))
Using the property a^(m+n) = a^m * a^n, we separate the terms:
f(x) = 5 * 2^x * 2^(x/3)
This can be written as:
f(x) = 5 * 2^x * ∛2^x
Or equivalently,
f(x) = 5 * (2 ∛2)^x
Thus, the simplification f(x) = 5 ∛16^x = 5 (2 ∛2)^x is correct. The initial function and the simplified form are mathematically equivalent, showcasing the accurate application of exponent rules and simplification techniques. This detailed breakdown ensures that each step is clear, providing a solid understanding of the process involved.
B) f(x) = 2.3 (8)^(1/2 x) = 2.3 (4)^x
Next, we will analyze the second exponential function: f(x) = 2.3 (8)^(1/2 x). The main task here is to simplify the term (8)^(1/2 x). To do this, we first express the base, 8, as a power of 2. We know that 8 = 2^3. Substituting this into the function, we get:
f(x) = 2.3 (23)(1/2 x)
Using the exponent rule (am)n = a^(m*n), we multiply the exponents:
f(x) = 2.3 * 2^(3 * (1/2 x))
This simplifies to:
f(x) = 2.3 * 2^(3x/2)
Now, let's compare this simplified form with the proposed simplification 2.3 (4)^x. To determine if these are equivalent, we need to express 4 as a power of 2, which is 4 = 2^2. So, 2.3 (4)^x can be written as:
2.3 * (22)x
Again, using the rule (am)n = a^(m*n), we get:
2.3 * 2^(2x)
Comparing this with our earlier simplified form 2.3 * 2^(3x/2), we see that the exponents are different (3x/2 vs. 2x). Therefore, the simplification f(x) = 2.3 (8)^(1/2 x) = 2.3 (4)^x is incorrect. The function 2.3 * 2^(3x/2) and 2.3 * 2^(2x) represent different exponential behaviors, highlighting the importance of careful simplification.
C) f(x) = 81^(x/4) = 3^x
Now, let's consider the third exponential function: f(x) = 81^(x/4). Simplifying this function involves expressing the base, 81, as a power of a smaller number. We know that 81 can be written as 3^4. Therefore, the function becomes:
f(x) = (34)(x/4)
Using the exponent rule (am)n = a^(m*n), we multiply the exponents:
f(x) = 3^(4 * (x/4))
This simplifies to:
f(x) = 3^x
Comparing this simplified form with the proposed simplification 3^x, we see that they are identical. This means the simplification f(x) = 81^(x/4) = 3^x is correct. The accurate transformation of 81^(x/4) to 3^x showcases a clear understanding of exponential properties and simplification techniques. This result demonstrates how expressing numbers in terms of their prime factors can significantly simplify exponential functions.
D) f(x) = (3/4) √27^x = (3/4) (3 √3)^x
Finally, we examine the fourth exponential function: f(x) = (3/4) √27^x. To simplify this function, we need to express the square root and the number 27 in terms of their prime factors. We know that 27 can be written as 3^3. Therefore, the function becomes:
f(x) = (3/4) √(33)x
Using the exponent rule (am)n = a^(m*n), we get:
f(x) = (3/4) √3^(3x)
The square root can be expressed as an exponent of 1/2. Thus, the function can be further rewritten as:
f(x) = (3/4) (3(3x))(1/2)
Applying the property of exponents (am)n = a^(m*n) again, we multiply the exponents:
f(x) = (3/4) * 3^(3x/2)
To simplify this further, we can express 3^(3x/2) as 3^(x * 3/2), which is equivalent to 3^(x * (1 + 1/2)). Splitting the exponent, we get:
f(x) = (3/4) * 3^(x * (1 + 1/2))
Using the property a^(m+n) = a^m * a^n, we separate the terms:
f(x) = (3/4) * 3^x * 3^(x/2)
This can be written as:
f(x) = (3/4) * 3^x * √3^x
Or equivalently,
f(x) = (3/4) * (3 √3)^x
Thus, the simplification f(x) = (3/4) √27^x = (3/4) (3 √3)^x is correct. The steps taken to transform the original function into its simplified form demonstrate a solid understanding of exponential rules and simplification techniques. This comprehensive analysis highlights the importance of breaking down complex expressions into their prime factors and applying the appropriate exponent rules.
Conclusion
In summary, we have meticulously analyzed four exponential functions and their simplified forms. Through this detailed exploration, we have identified the correct simplifications and explained the underlying mathematical principles. Options A, C, and D were simplified correctly, while option B was not. Mastering the simplification of exponential functions is crucial for various mathematical applications, and this guide aims to provide a thorough understanding of the process. By understanding these concepts, you will be better equipped to tackle more complex mathematical problems and real-world applications involving exponential growth and decay.
