Equivalent Expression Of (x^(1/4)y^16)^(1/2): Step-by-Step

Understanding and simplifying expressions with fractional exponents can sometimes feel like navigating a mathematical maze. However, by applying the fundamental rules of exponents, we can break down even complex-looking expressions into simpler, more manageable forms. This article will guide you through simplifying the expression (x^(1/4) * y¹⁶⁾(1/2), explaining each step in detail so you can confidently tackle similar problems. Whether you're a student looking to ace your next math test or someone who enjoys the elegance of mathematical simplification, this guide is for you. Let’s dive in and unlock the secrets of this expression!

Breaking Down the Expression

When you're faced with an expression like (x^(1/4) * y¹⁶⁾(1/2), it might seem daunting at first glance. But don't worry! The key is to remember the fundamental rules of exponents, especially the power of a product rule and the power of a power rule. Let's break down the expression step by step to make it easier to understand. The expression we need to simplify is (x^(1/4) * y¹⁶⁾(1/2). This expression involves fractional exponents and the product of variables raised to powers, all enclosed within parentheses and raised to another power. Our goal is to simplify this expression using the rules of exponents. First, let's identify the core components: We have the variables x and y, each raised to a power. These terms are multiplied together, and the entire product is raised to the power of 1/2. To simplify, we'll apply the power of a product rule, which states that (ab)^n = a^n * b^n. This means we need to distribute the outer exponent (1/2) to each term inside the parentheses. This step is crucial because it allows us to deal with each variable separately, making the simplification process much clearer. By applying the power of a product rule, we transform the single complex expression into a series of simpler expressions that are easier to manage. This is a common strategy in algebra – breaking down a problem into smaller, more manageable parts. By doing so, we avoid getting overwhelmed and can focus on applying the appropriate rules to each part.

Applying the Power of a Product Rule

To simplify the expression (x^(1/4) * y¹⁶⁾(1/2), the first critical step is to apply the power of a product rule. This rule is a cornerstone of exponent manipulation and states that when a product of terms is raised to a power, you can distribute the power to each term individually. In mathematical terms, it's expressed as (ab)^n = a^n * b^n. This rule is particularly useful when dealing with expressions involving both multiplication and exponents. Applying this rule to our expression, we get: (x^(1/4) * y¹⁶⁾(1/2) = x^(1/4)(1/2) * y¹⁶(1/2). What we've done here is distribute the exponent of 1/2 to both x^(1/4) and y^16. This effectively separates the expression into two parts, each of which can be simplified independently. By applying the power of a product rule, we've transformed the original expression into a form where we can now focus on simplifying each term separately. This is a common technique in algebra – breaking down complex problems into smaller, more manageable parts. This separation allows us to apply other exponent rules more easily and reduces the chances of making errors. Now, we have two separate terms, x^(1/4)(1/2) and y¹⁶(1/2), each involving a power raised to another power. To simplify these, we'll need to use another important rule of exponents, which we'll discuss in the next section. Understanding and applying the power of a product rule is essential for simplifying expressions like this. It’s a fundamental tool in algebra and a stepping stone to more complex mathematical manipulations.

Simplifying with the Power of a Power Rule

After applying the power of a product rule, our expression looks like this: x^(1/4)(1/2) * y¹⁶(1/2). Now, we need to simplify each term individually, and for that, we'll use the power of a power rule. This rule states that when you raise a power to another power, you multiply the exponents. Mathematically, this is represented as (a^m)n = a^(m*n). This rule is incredibly useful for simplifying expressions where exponents are nested, like in our current situation. Let's apply the power of a power rule to the first term, x^(1/4)(1/2). According to the rule, we multiply the exponents 1/4 and 1/2: x^(1/4 * 1/2) = x^(1/8). So, the first term simplifies to x^(1/8). Now, let's apply the same rule to the second term, y¹⁶(1/2). Here, we multiply the exponents 16 and 1/2: y^(16 * 1/2) = y^8. Thus, the second term simplifies to y^8. By applying the power of a power rule to both terms, we've significantly simplified the expression. The nested exponents have been resolved, and we now have a much cleaner representation of the original expression. Understanding and correctly applying the power of a power rule is crucial for simplifying expressions with exponents. It’s a fundamental concept in algebra and a key skill for solving more complex mathematical problems. Now that we've simplified each term individually, we can combine them to get the final simplified expression.

Combining the Simplified Terms

Having simplified each term individually using the power of a power rule, we now have x^(1/8) and y^8. The final step is to combine these simplified terms to obtain the equivalent expression for the original problem, which was (x^(1/4) * y¹⁶⁾(1/2). Since we initially separated the expression using the power of a product rule, we now simply multiply the simplified terms together. This gives us: x^(1/8) * y^8. This is the simplified form of the original expression. We started with a seemingly complex expression and, by systematically applying the rules of exponents, have arrived at a much simpler form. This final expression, x^(1/8) * y^8, is equivalent to the original expression (x^(1/4) * y¹⁶⁾(1/2). It’s important to double-check your work at this stage to ensure that all simplifications have been done correctly and that the final expression is indeed equivalent to the original. This involves reviewing each step, from applying the power of a product rule to using the power of a power rule, and ensuring that no errors were made in the calculations. Combining the simplified terms is the culmination of all the previous steps. It demonstrates the power of breaking down a complex problem into smaller parts and applying the appropriate rules to each part. By doing so, we can simplify even the most challenging expressions and arrive at a clear and concise solution.

Final Answer and Conclusion

After carefully applying the power of a product rule and the power of a power rule, we've successfully simplified the expression (x^(1/4) * y¹⁶⁾(1/2). The final simplified expression is x^(1/8) * y^8. This corresponds to option B in the multiple-choice options provided at the beginning. To recap, we started by recognizing the structure of the expression and identifying the key rules of exponents that would help us simplify it. We first applied the power of a product rule to distribute the outer exponent to each term inside the parentheses. This allowed us to separate the expression into two simpler parts. Next, we used the power of a power rule to simplify each term individually, multiplying the exponents as needed. Finally, we combined the simplified terms to arrive at the final answer. Understanding and applying these rules of exponents is crucial for simplifying algebraic expressions. It’s a fundamental skill in mathematics and is used extensively in various branches of math and science. By mastering these rules, you can confidently tackle more complex problems and gain a deeper understanding of mathematical concepts. Remember, practice makes perfect. The more you work with these rules, the more comfortable you'll become with them. So, keep practicing, and don't hesitate to review the steps we've covered in this guide whenever you need a refresher.

For further learning on exponents and their properties, you can visit Khan Academy's Exponents and Polynomials section.