Simplifying Exponential Expressions Equivalent To (4^(5/4) * 4^(1/4))/(4^(1/2))^(1/2)

In this article, we will delve into simplifying exponential expressions, focusing on the specific example: ${\left(\frac{4^{\frac{5}{4}} \cdot 4^{\frac{1}{4}}}{4^{\frac{1}{2}}}\right)^{\frac{1}{2}}}$. This problem requires a solid understanding of exponent rules, including the product of powers, quotient of powers, and power of a power rules. By breaking down the expression step-by-step, we can arrive at the correct answer and gain a deeper appreciation for manipulating exponents. This guide aims to provide a comprehensive explanation, ensuring clarity for anyone looking to master these concepts. Let's explore each step in detail, making the process both understandable and engaging. Exponential expressions might seem daunting at first, but with the right approach, they can be simplified systematically. Our focus will be on applying the fundamental rules of exponents to transform the given expression into its simplest form. This journey will not only help solve this specific problem but also equip you with the skills to tackle similar challenges with confidence. Whether you are a student learning these concepts for the first time or someone looking to refresh your knowledge, this guide will serve as a valuable resource. We will use clear and concise explanations, breaking down each rule and its application in the context of our problem. By the end of this article, you will have a thorough understanding of how to simplify complex exponential expressions effectively.

Understanding the Problem

Before we dive into the solution, let's clearly state the problem: Simplify the expression ${\left(\frac{4^{\frac{5}{4}} \cdot 4^{\frac{1}{4}}}{4^{\frac{1}{2}}}\right)^{\frac{1}{2}}}$. This requires us to apply the fundamental rules of exponents systematically. To effectively simplify this expression, we must first recognize the operations involved and the order in which they should be applied. The expression involves fractional exponents, multiplication, division, and the power of a power. Each of these components requires a specific rule to be applied correctly. The first step is to simplify the terms within the parentheses, which includes dealing with the multiplication and division of exponents with the same base. Once we have simplified the expression inside the parentheses, we can then apply the outer exponent. This methodical approach ensures that we handle each component of the expression correctly, leading us to the simplified form. It's crucial to remember that exponents represent repeated multiplication, and the rules governing them are derived from this fundamental concept. By understanding the underlying principles, we can avoid common mistakes and approach simplification with confidence. The problem at hand is not just about finding the correct answer but also about mastering the process of simplifying exponential expressions. This skill is invaluable in various areas of mathematics and science. By breaking down the problem into smaller, manageable steps, we make it easier to understand and solve. This approach also helps in identifying any potential errors and correcting them along the way. Let's embark on this journey of simplification, ensuring that each step is clear and logical, ultimately leading us to the solution.

Step-by-Step Solution

Step 1 Apply the Product of Powers Rule

The first step in simplifying the expression is to apply the product of powers rule to the numerator. This rule states that when multiplying exponential expressions with the same base, you add the exponents. In our case, we have ${4^{\frac{5}{4}} \cdot 4^{\frac{1}{4}}}$. Applying the rule, we add the exponents ${\frac{5}{4}}$ and ${\frac{1}{4}}$:

${4^{\frac{5}{4}} \cdot 4^{\frac{1}{4}} = 4^{\frac{5}{4} + \frac{1}{4}} = 4^{\frac{6}{4}}}$

So, the numerator simplifies to ${4^{\frac{6}{4}}}$ or ${4^{\frac{3}{2}}}$. This step is crucial because it combines two terms into one, making the expression easier to manage. The product of powers rule is a fundamental concept in exponent manipulation, and its correct application is essential for simplifying expressions effectively. This rule is not just a mathematical shortcut; it's a reflection of the underlying principle of repeated multiplication. When we multiply ${4^{\frac{5}{4}}}$ by ${4^{\frac{1}{4}}}$, we are essentially combining the fractional powers, which results in a single power that is the sum of the individual powers. Understanding this principle helps in memorizing the rule and applying it correctly in various contexts. The ability to apply this rule efficiently can significantly reduce the complexity of exponential expressions and make them easier to work with. Now that we have simplified the numerator, we can move on to the next step, which involves dealing with the division.

Step 2 Apply the Quotient of Powers Rule

Now that we've simplified the numerator to (4^{\frac{3}{2}}, we can apply the quotient of powers rule. This rule states that when dividing exponential expressions with the same base, you subtract the exponents. Our expression now looks like this:

${\frac{4^{\frac{3}{2}}}{4^{\frac{1}{2}}}}$

Applying the quotient of powers rule, we subtract the exponents ${\frac{1}{2}}$ from ${\frac{3}{2}}$:

${\frac{4^{\frac{3}{2}}}{4^{\frac{1}{2}}} = 4^{\frac{3}{2} - \frac{1}{2}} = 4^{\frac{2}{2}} = 4^1}$

Thus, the expression inside the parentheses simplifies to ${4^1}$, which is simply 4. The quotient of powers rule, much like the product of powers rule, is a cornerstone of exponent manipulation. It's based on the principle that division is the inverse operation of multiplication. When we divide exponential expressions with the same base, we are essentially canceling out some of the repeated multiplications, which results in subtracting the exponents. This rule is not only useful in simplifying expressions but also in solving equations involving exponents. The ability to quickly apply this rule can significantly streamline the simplification process and reduce the likelihood of errors. By understanding the underlying logic behind the rule, we can apply it with greater confidence and accuracy. Now that we have simplified the expression inside the parentheses to 4, we can proceed to the final step, which involves applying the outer exponent.

Step 3 Apply the Power of a Power Rule

After simplifying the expression inside the parentheses to 4, we now have to deal with the outer exponent of ${\frac{1}{2}}$. This means we need to apply the power of a power rule, which states that when raising an exponential expression to a power, you multiply the exponents. Our expression is now:

${(4)^{\frac{1}{2}}}$

Here, we have ${4^1}$ raised to the power of ${\frac{1}{2}}$. Multiplying the exponents, we get:

${4^{1 \cdot \frac{1}{2}} = 4^{\frac{1}{2}}}$

Now, we recognize that ${4^{\frac{1}{2}}}$ is the same as the square root of 4:

${4^{\frac{1}{2}} = \sqrt{4} = 2}$

Therefore, the simplified form of the original expression is 2. The power of a power rule is another fundamental concept in exponent manipulation. It's based on the idea that raising an exponential expression to a power is equivalent to repeated exponentiation. This rule is particularly useful when dealing with nested exponents, as it allows us to simplify the expression by multiplying the exponents together. Understanding this rule is crucial for simplifying complex expressions and solving equations involving exponents. In our case, applying the power of a power rule allowed us to transform the expression into a more familiar form, which we could then easily simplify. This step highlights the importance of recognizing the different forms of exponential expressions and knowing how to convert between them. With this final step, we have successfully simplified the original expression and arrived at the solution. Let's now review the entire process and the answer.

Final Answer

After carefully applying the exponent rules step-by-step, we have found that the expression ${\left(\frac{4^{\frac{5}{4}} \cdot 4^{\frac{1}{4}}}{4^{\frac{1}{2}}}\right)^{\frac{1}{2}}}$ simplifies to 2. This corresponds to option C in the given choices. Let's recap the steps we took to arrive at this answer: First, we applied the product of powers rule to the numerator, combining the exponents. Second, we used the quotient of powers rule to simplify the fraction. Finally, we applied the power of a power rule and evaluated the resulting expression. Each of these steps is crucial in simplifying exponential expressions, and mastering them will greatly enhance your ability to solve similar problems. This problem serves as a great example of how complex-looking expressions can be simplified by systematically applying the rules of exponents. The key is to break down the expression into smaller, manageable steps and to apply the appropriate rule at each step. By following this approach, you can confidently tackle a wide range of exponential problems. The final answer of 2 is not just a number; it represents the culmination of a series of logical steps and the correct application of mathematical principles. Understanding the process is as important as finding the answer, as it allows you to generalize the solution and apply it to other problems. This journey through simplification has not only given us the answer but also reinforced our understanding of exponent rules and their applications. With this knowledge, you are well-equipped to handle future challenges involving exponential expressions.

Conclusion

In conclusion, simplifying exponential expressions involves a methodical application of exponent rules. We successfully simplified ${\left(\frac{4^{\frac{5}{4}} \cdot 4^{\frac{1}{4}}}{4^{\frac{1}{2}}}\right)^{\frac{1}{2}}}$ to 2 by using the product of powers, quotient of powers, and power of a power rules. Mastering these rules is essential for simplifying various mathematical expressions. This exercise demonstrates the power of breaking down complex problems into smaller, manageable steps. By understanding the underlying principles and applying the rules correctly, we can transform seemingly daunting expressions into simpler forms. The journey through simplification is not just about finding the answer; it's about developing a deeper understanding of mathematical concepts and enhancing problem-solving skills. The ability to simplify exponential expressions is a valuable asset in various fields, including mathematics, science, and engineering. It allows us to manipulate equations, solve problems, and gain insights into the relationships between different quantities. This article has provided a comprehensive guide to simplifying the given expression, but the principles and techniques discussed can be applied to a wide range of similar problems. By practicing and applying these concepts, you can build confidence and proficiency in simplifying exponential expressions. The key is to approach each problem systematically, identify the relevant rules, and apply them carefully. With dedication and practice, you can master the art of simplifying exponential expressions and unlock the power of mathematics.