Expressing 2x^2y^4(⁴√3x^3y) As An Entire Radical

In the realm of algebra, expressing radicals in their simplest forms is a fundamental skill. Often, we encounter expressions that combine terms inside and outside a radical. To simplify such expressions and make them easier to work with, we can express them as entire radicals. An entire radical is a radical expression where all terms are brought under the radical sign. This process often involves understanding how to manipulate exponents and indices of radicals. In this article, we will delve into expressing the given expression, $2x^2y^4(\sqrt[4]{3x^3y})$, as an entire radical, providing a step-by-step breakdown to ensure clarity and comprehension. This technique is crucial not only for simplifying expressions but also for performing more complex algebraic operations like comparing and combining radicals.

Understanding Entire Radicals

Before diving into the specific example, let's first clarify what an entire radical is and why it's a useful concept in algebra. An entire radical is a radical expression in which all factors are included under the radical sign. In contrast, a mixed radical has factors both inside and outside the radical. Converting a mixed radical to an entire radical often simplifies the expression, making it easier to compare, combine, or further manipulate in algebraic equations. The transformation involves raising the factors outside the radical to the power of the radical's index and then multiplying them by the factors already inside the radical. This process hinges on the properties of exponents and radicals, where the nth root of a number raised to the nth power is the number itself, i.e., $(\sqrt[n]{a})^n = a$. By understanding this principle, we can effectively move terms in and out of radicals, which is a critical skill in algebraic simplification.

Step-by-Step Conversion

Step 1: Identify the Expression

The expression we aim to convert to an entire radical is $2x^2y^4(\sqrt[4]{3x^3y})$. Here, we have a mix of terms outside the radical ($2x^2y^4$) and terms inside the fourth root radical ($\sqrt[4]{3x^3y}$). The goal is to bring the outside terms under the fourth root. To accomplish this, we need to understand how the index of the radical affects the terms being brought inside. The index, which is 4 in this case, dictates the power to which the outside terms must be raised before they can be included under the radical. This is because taking the fourth root is the inverse operation of raising to the fourth power, and we are essentially reversing the process of simplifying a radical. By adhering to this principle, we ensure that the value of the expression remains unchanged while its form is altered.

Step 2: Raise Outside Terms to the Power of the Index

The next step involves raising the terms outside the radical, which are $2$, $x^2$, and $y^4$, to the power of the index of the radical, which is 4. This means we calculate $2^4$, $(x^2)^4$, and $(y^4)^4$. Recall that when raising a power to a power, we multiply the exponents. Therefore, $(x^2)^4$ becomes $x^{2*4} = x^8$, and $(y^4)^4$ becomes $y^{4*4} = y^{16}$. Additionally, $2^4$ is $2 * 2 * 2 * 2 = 16$. These calculations are crucial because they determine how the outside terms will be modified when brought under the radical. Raising these terms to the appropriate power ensures that when we take the fourth root later (if we were to simplify back), we would return to the original expression. This step highlights the interplay between exponents and radicals, a fundamental concept in algebraic manipulations.

Step 3: Multiply Inside Terms by the Result

Now that we've raised the outside terms to the power of the index, we can bring them under the radical. We do this by multiplying the results from Step 2 by the terms already inside the radical, which are $3x^3y$. So, we multiply $16 * x^8 * y^{16}$ by $3x^3y$. This results in $(16 * 3) * (x^8 * x^3) * (y^{16} * y)$. When multiplying terms with the same base, we add their exponents. Thus, $x^8 * x^3$ becomes $x^{8+3} = x^{11}$, and $y^{16} * y$ (where $y$ is equivalent to $y^1$) becomes $y^{16+1} = y^{17}$. Also, $16 * 3$ equals 48. Therefore, the terms under the radical become $48x^{11}y^{17}$. This step effectively combines all factors under a single radical, fulfilling the requirement of an entire radical form.

Step 4: Write the Entire Radical

Finally, we write the entire expression under the radical. The result from Step 3, $48x^{11}y^{17}$, now goes under the fourth root. Thus, the entire radical expression is $\sqrt[4]{48x^{11}y^{17}}$. This completes the conversion of the mixed radical to an entire radical. The expression is now in a simplified form where all factors are under the same radical, making it easier to compare with other radicals or to perform further algebraic operations. This final step demonstrates the power of manipulating expressions using exponent and radical rules to achieve a more streamlined form.

Final Answer

Therefore, the expression $2x^2y^4(\sqrt[4]{3x^3y})$ expressed as an entire radical is $\sqrt[4]{48x^{11}y^{17}}$.

Conclusion

In this discussion, we successfully converted the expression $2x^2y^4(\sqrt[4]{3x^3y})$ into an entire radical, which is $\sqrt[4]{48x^{11}y^{17}}$. The process involved understanding how to manipulate terms both inside and outside the radical, raising the outside terms to the power of the radical's index, and then combining them under the radical. This technique is not only essential for simplifying expressions but also for making them easier to compare and combine. Mastering these skills allows for more efficient and accurate manipulation of algebraic expressions, a crucial aspect of mathematical problem-solving. Understanding entire radicals and their manipulation forms a cornerstone in algebra, enabling students and professionals alike to tackle complex problems with confidence and clarity. The ability to convert between mixed and entire radicals is a powerful tool in simplifying and solving a wide range of mathematical problems, making it an indispensable skill in any mathematical toolkit.