Problem 82 - Algebra 2

Question

Simplify each radical expression. Use absolute value symbols as needed. $$ \sqrt[4]{16 x^{36} y^{96}} $$

Step-by-Step Solution

Answer

The simplified radical expression is \(2x^9y^{24}\).

Step 1: Break Down the Radical Expression

Here, we break down the fourth root into its individual components: \(16\), \(x^{36}\), and \(y^{96}\).

Step 2: Simplify Each Component

We can simplify each component individually. The fourth root of \(16\) is \(2\), the exponent of \(x^{36}\) divided by the root \(4\) is \(9\), and the exponent of \(y^{96}\) divided by the root \(4\) is \(24\). Note, that no absolute values are needed because all the resulting powers are even.

Step 3: Combine and Simplify

Now we combine the results in a multiplication. So, our simplified expression is \(2x^9y^{24}\).

Key Concepts

Exponents and Radicals

Exponents and radicals are fundamental concepts in algebra that are used to express powers and roots of numbers. When dealing with roots, especially, it is important to understand how they relate to exponents. For example, the square root of a number can be represented as the number raised to the power of one-half. Similarly, a fourth root, which is our focus here, is a number raised to the power of one-fourth.
Understanding how to manipulate exponents and radicals is crucial because it allows you to simplify expressions more effectively. For instance, if you have an expression under a radical, you can often use the property \(\sqrt[n]{a^m} = a^{m/n}\) to simplify it. This is exactly what we do when breaking down expressions like \(\sqrt[4]{16x^{36}y^{96}}\). By transforming roots into fractional exponents, you make calculations straightforward and enable the simplification of complex algebraic expressions.

Fourth Roots

Fourth roots are a specific kind of radical expression represented by \(\sqrt[4]{\cdot}\). The primary function of the fourth root is to return a number that, when raised to the power of 4, gives the original number. For example, \(\sqrt[4]{16}\) equals 2 because \(2^4 = 16\).
When simplifying fourth roots, each component within the radical can often be dealt with separately, especially if they are products. This is because the fourth root of a product is the product of the fourth roots:

• \(\sqrt[4]{abcd} = \sqrt[4]{a} \times \sqrt[4]{b} \times \sqrt[4]{c} \times \sqrt[4]{d}\)

Using this property, you can simplify more complex expressions by assessing each part individually. In the expression \(\sqrt[4]{16x^{36}y^{96}}\), we simplified each component independently before combining them.

Polynomial Simplification

Polynomial simplification involves reducing expressions to their simplest form by combining like terms, factoring, or reducing elements with common factors. When given an expression such as \(\sqrt[4]{16x^{36}y^{96}}\), the goal is to express it as simply as possible, eliminating any roots or exponents where feasible.
Breaking down and simplifying exponents can significantly reduce the complexity. Here:

• The fourth root of \(x^{36}\) gives us \(x^9\) since 36 divided by 4 is 9.
• The fourth root of \(y^{96}\) yields \(y^{24}\) because 96 divided by 4 is 24.

No absolute value symbols were necessary here because all resulting exponents were even, ensuring all simplified variables remained positive. This exemplifies how understanding the structure of polynomials can streamline their simplification. By applying these principles, we transformed the original radical expression into a more manageable algebraic expression \(2x^9y^{24}\).