Problem 108 - College Algebra Essentials

Question

In Exercises \(101-108\), simplify by reducing the index of the radical. $$\sqrt[12]{x^{4} y^{8}}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(x^{1/3}y^{2/3}\).

Step 1: Break down the variables

Firstly, break down the variables into powers that divide evenly by the index of the radical. Here \(x^{4}\) and \(y^{8}\) divide evenly by 12. So, we can rewrite the expression as \(\sqrt[12]{{(x^{1})^{4}(y^{2})^{4}}}\).

Step 2: Simplify the expression

Next, apply the property of radicals which states that \(\sqrt[n]{{(a^{m})^{k}}}\) can be simplified to \(a^{mk/n}\). Therefore \(x^{4/12}y^{8/12}\) simplifies to \(x^{1/3}y^{2/3}\).

Step 3: Write the final answer

Finally, write down the simplified expression \(x^{1/3}y^{2/3}\). This is the expression where the index of the radical has been reduced to its lowest form.

Key Concepts

Understanding Powers and Roots

Simplifying radicals often involves dealing with powers and roots. A root is represented by a radical sign, and the index of the radical tells us which root we are dealing with. For example, the square root has an index of 2, and a cube root has an index of 3. In our exercise, we have a 12th root, showing that the index is 12.

Powers, on the other hand, are exponents. In the expression under the radical, such as \(x^4 y^8\), the \(4\) and \(8\) are powers of \(x\) and \(y\) respectively. To simplify, we look for combinations of these powers that can fit into the root indicated by the index. By breaking down these powers to fit the index, simplifying becomes more manageable.

Exploring Rational Exponents

Rational exponents are another way to represent roots. Instead of using radical signs, we can use fractional exponents. For instance, the expression \(\sqrt[n]{a}\) can be rewritten as \(a^{1/n}\). This shows the connection between exponents and roots in a straightforward manner.

In our exercise, we transformed the expression \(\sqrt[12]{x^4 y^8}\) into \(x^{4/12} y^{8/12}\), using rational exponents. These fractions simplify to \(x^{1/3} y^{2/3}\). Working with rational exponents makes it easier to apply algebraic rules and simplify expressions. It's all about rewriting the expression in a form that is easier to manage.

Reducing the Index of Radicals

Reducing the index of radicals is about simplifying the expression to its most basic form. The index tells us the degree of the root, and our goal is to reduce it.

To do this, we divide the exponents of the variables by the index of the radical. In our example, the index was 12, and we divided the powers of each variable with 12, resulting in \(x^{1/3} y^{2/3}\). This process makes the expression simpler and more concise.

By rewriting radicals with lower indices or even as rational exponents, we can better understand the relationship between different components of an expression. It's a handy tool for anyone working with algebraic expressions to achieve simplicity and clarity.