Solving The Expression: (4²)^(1/4) Explained

Hey math enthusiasts! Let's dive into a neat little problem that often pops up in the world of exponents and roots. We're going to break down how to solve the expression $\left(4^2\right)^{1 / 4}$. It's all about understanding the rules of exponents and applying them step by step. Don't worry, it's not as scary as it looks. By the end of this, you'll be comfortable tackling similar problems! First things first, let's look at the expression itself. We have $\left(4^2\right)^{1 / 4}$. This means we need to deal with a power raised to another power. The key concept here is understanding how exponents interact with each other.

Understanding the Basics: Exponents and Roots

Before we jump into the solution, let's refresh our memory on some key principles. The expression $4^2$ means 4 multiplied by itself, which is $4 \times 4 = 16$. So, the first part of our problem is pretty straightforward. Then, we have the exponent $1/4$. This fractional exponent is a way of representing a root. Specifically, an exponent of $1/4$ means we are taking the fourth root. The fourth root of a number is the value that, when raised to the power of 4, gives you the original number. For example, the fourth root of 16 is 2 because $2^4 = 2 \times 2 \times 2 \times 2 = 16$. Now, armed with this knowledge, we are ready to proceed with solving $\left(4^2\right)^{1 / 4}$. Keep in mind that when we have a power raised to another power, we multiply the exponents. In our case, this means we will multiply 2 by $1/4$. The goal is to make the process as clear and easy to grasp as possible. We will explain the rationale behind each step, ensuring you understand not just how to solve the problem, but also why the solution works the way it does. This approach will equip you with a solid foundation for tackling more complex mathematical problems in the future. Remember, math is all about understanding the 'why' behind the 'how'. So, let's get started!

Step-by-Step Solution

Let's get down to business and break down how to solve $\left(4^2\right)^{1 / 4}$.

1. Simplify the Inner Expression: First, let's deal with the term inside the parentheses, which is $4^2$. As we mentioned earlier, $4^2$ means $4 \times 4$, which equals 16. So, our expression now becomes $16^{1 / 4}$.
2. Apply the Fractional Exponent: The exponent $1/4$ indicates that we need to find the fourth root of 16. Think of it this way: what number, when multiplied by itself four times, gives us 16? As we saw before, the answer is 2, since $2 \times 2 \times 2 \times 2 = 16$. So, $16^{1 / 4} = 2$. Therefore, the solution to the original expression $\left(4^2\right)^{1 / 4}$ is 2. The process is straightforward when broken down into manageable steps, highlighting the beauty and simplicity of mathematical operations. It's a journey of transforming a complex-looking expression into a simple, elegant solution. Through this step-by-step approach, we not only arrive at the correct answer but also reinforce the fundamental principles of exponents and roots. This method provides a clear understanding and builds confidence to tackle any similar problem with ease. The secret lies in breaking down the problem into smaller, easier-to-manage parts and then applying the appropriate mathematical rules. Once you grasp these basics, you're well-equipped to handle more complex mathematical challenges.
3. Final Answer: So, the value of $\left(4^2\right)^{1 / 4}$ is 2. This matches option A in the multiple-choice question.

Why This Works: Exponent Rules Explained

Let's clarify why we can solve this using exponents rules. There's a fundamental rule that comes into play here: when you raise a power to another power, you multiply the exponents. In our expression, $\left(4^2\right)^{1 / 4}$, the exponent rule states that we multiply the exponent 2 by the exponent $1/4$. Essentially, the rule tells us to multiply the powers together: $2 \times (1/4) = 1/2$. So, if we apply this rule first, the expression becomes $4^{1/2}$. But note that we followed a different, easier path: First, simplify within the parentheses, then calculate the root. Understanding this rule helps simplify many problems. This means we're looking for the square root of 4, which is 2. This is just another way to get to the same answer. Both methods are valid, but understanding the rule of multiplying exponents provides a deeper insight into how exponents behave. The key takeaway is understanding how the rules of exponents simplify complex-looking expressions into more manageable forms. Recognizing these patterns and rules allows for efficient and accurate solutions in mathematics. Remember, the elegance of mathematics lies in its simplicity and the interconnectedness of its principles. Mastering these rules not only allows you to solve problems quickly but also builds a strong foundation for more advanced mathematical concepts.

Conclusion: Mastering Exponents and Roots

So, there you have it! We've successfully solved the expression $\left(4^2\right)^{1 / 4}$, and the answer is 2. By breaking down the problem step by step, we've demonstrated how to apply the rules of exponents and roots. Understanding these concepts is crucial for various mathematical problems. Keep practicing, and you will become more comfortable with these types of expressions. The more you work with exponents and roots, the more familiar you will become with their properties, making solving similar problems a breeze. Remember, math is a skill that improves with practice, so don't be afraid to try different problems and approaches. Each problem solved is a step forward in your mathematical journey. The key is to stay curious and keep exploring. With each expression, you reinforce your understanding and build confidence in your problem-solving abilities. Stay curious, keep practicing, and enjoy the process of learning. The world of mathematics is vast and full of exciting discoveries!

For further exploration, you can visit Khan Academy.