Rewriting Exponential Functions: A Step-by-Step Guide

Hey everyone! Today, we're diving into the world of exponential functions and learning how to rewrite them using the awesome properties of exponents. This is super useful for simplifying expressions, understanding how functions behave, and tackling more complex problems. So, let's get started and break down the question: Rewrite the function $f(x)=7\left(\frac{1}{2}\right)^{3 x}$ using the properties of exponents. We'll go through the problem step-by-step and then look at the multiple-choice options to find the correct answer. This is an important concept in algebra, so pay close attention, folks!

Understanding the Basics: Exponential Functions

Before we jump into the problem, let's quickly recap what an exponential function is all about. An exponential function is a function where the variable appears in the exponent. The general form is $f(x) = a * b^x$, where:

• a is the initial value (the value of the function when x = 0).
• b is the base (the number being raised to the power of x). The base determines whether the function increases (b > 1) or decreases (0 < b < 1) as x increases.
• x is the exponent (the variable).

In our case, the given function is $f(x)=7\left(\frac{1}{2}\right)^{3 x}$. Here, a = 7 and b = (1/2)^{3x}. Our goal is to rewrite this function in a more simplified form, where we have a single base raised to the power of x. This involves using the power of a power property, which we'll explore shortly. Understanding these components is critical to manipulating and simplifying exponential functions effectively. It's like knowing the ingredients before you start cooking; it makes the whole process smoother and more understandable. Exponential functions are fundamental in many areas, including finance (compound interest), physics (radioactive decay), and computer science (algorithm analysis), making this skill a valuable one to master. So, keep this in mind as you see how we will approach this problem.

The Power of a Power Property

The key to solving this problem lies in understanding and applying the power of a power property. This property states that $(a^m)^n = a^{m*n}$. In simpler terms, when you have a power raised to another power, you multiply the exponents. This property allows us to simplify expressions and rewrite them in different, yet equivalent, forms. This is one of the most important rules when dealing with exponents, and it unlocks many possibilities for simplifying complex expressions. For example, if we have $(2^3)^2$, we can simplify it as $2^{3*2} = 2^6 = 64$. Now, let's relate this to the problem we have. We'll use this property to simplify the base of our exponential function. Pay close attention to how we do this.

We need to rewrite the given function $f(x)=7\left(\frac{1}{2}\right)^{3 x}$. Notice that we have a power raised to a power situation. The base is 1/2, and it's being raised to the power of 3x. We can rewrite 3x as 3 * x. This means that we can apply the power of a power property to simplify the base. It is worth noting, that sometimes the most difficult part of a problem is identifying the right property, so it’s important to practice and recognize patterns. Once we’ve done that, it’s usually smooth sailing. This will allow us to rewrite the function in a more compact and manageable form, which will help us match it with one of the provided options. So, keep the power of a power property in mind; it's the superhero of this problem. Remember that in mathematics, it is important to practice regularly.

Step-by-Step Solution

Alright, let's roll up our sleeves and solve this problem step-by-step. We'll start with the original function: $f(x)=7\left(\frac{1}{2}\right)^{3 x}$. Our aim is to rewrite this function in a form that has a single base raised to the power of x. Here's the breakdown:

1. Focus on the base: We have $(\frac{1}{2})^{3x}$.
2. Rewrite the exponent: We can rewrite $3x$ as $3 * x$. This allows us to use the power of a power property.
3. Apply the power of a power property: $(\frac{1}{2})^{3x}$ can be written as $((\frac{1}{2})^3)^x$.
4. Simplify the base: Calculate $(\frac{1}{2})^3$. This means $\frac{1}{2} * \frac{1}{2} * \frac{1}{2} = \frac{1}{8}$.
5. Rewrite the function: Now our function becomes $f(x)=7\left(\frac{1}{8}\right)^x$.

And there you have it, folks! We've successfully rewritten the function using the properties of exponents. We have transformed the original function into a more simplified form that is easier to analyze and work with. Note the order, it's very important to follow the steps to ensure that you are making the correct calculations. Also, by breaking down the problem into smaller steps, we've made the process more manageable and understandable. Remember, in mathematics, breaking down a problem is key to achieving success, and it is a strategy you can also use in your daily life. It really is a powerful skill. Now let's move on and compare our result with the multiple-choice options. You've done a great job!

Matching with the Options

Now that we've successfully rewritten the function, let's match our result with the given multiple-choice options. Remember, our simplified function is: $f(x)=7\left(\frac{1}{8}\right)^x$. Let's go through the choices:

• A. $f(x)=7\left(\frac{3}{6}\right)^x$: This doesn't match our simplified function. The base is incorrect.
• B. $f(x)=7\left(\frac{1}{8}\right)^x$: Bingo! This matches our simplified function perfectly.
• C. $f(x)=21\left(\frac{3}{6}\right)^x$: This doesn't match. The coefficient (the number multiplying the exponential term) is incorrect, and the base is also wrong.
• D. $f(x)=343\left(\frac{1}{8}\right)^x$: This doesn't match either. Both the coefficient and the base are incorrect.

So, the correct answer is B. $f(x)=7\left(\frac{1}{8}\right)^x$. Great job sticking with it! This question highlights the importance of understanding and correctly applying exponent properties, particularly the power of a power rule. Being able to manipulate expressions and rewrite them in different forms is a crucial skill in algebra and beyond. This ability allows you to solve a wider range of problems and gain a deeper understanding of mathematical concepts. Keep practicing, and you'll become a pro in no time! Also, you might find it useful to review the properties of exponents. Don’t be afraid to consult your textbook or online resources if you need a refresher.

Conclusion: Mastering Exponents

Congratulations, guys! You've successfully rewritten the exponential function and identified the correct answer. We've seen how important it is to understand the properties of exponents, especially the power of a power rule, to simplify and manipulate expressions. This is a fundamental concept in algebra and forms the foundation for more advanced topics in mathematics. Remember, practice is key to mastering these concepts. Work through similar problems, review the properties of exponents regularly, and don't be afraid to ask for help when you need it. By doing so, you'll build a strong foundation in mathematics and gain the confidence to tackle any problem that comes your way.

In summary, we've covered:

• Understanding exponential functions and their components.
• Applying the power of a power property.
• Rewriting the given function in a simplified form.
• Matching the rewritten function with the correct multiple-choice option.

Keep up the great work, and happy learning! I hope this explanation was helpful. If you have any questions or want to try some more examples, feel free to ask! Thanks for joining me today. Keep practicing and applying these concepts, and you'll become a master of exponents in no time. See you in the next lesson!