Problem 104 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

\(\sqrt[4]{x^{12}} = x^{3}\)

Step 1: Identify the Index and Radicand

In the radical \(\sqrt[4]{x^{12}}\), '4' is the index and \(x^{12}\) is the radicand.

Step 2: Apply the Radical Property

We know that \(\sqrt[n]{a^n} = a\). In this case, we can view the exponent of the radicand (12) as '4 * 3'. Therefore, the radicand \(x^{12}\) becomes \(x^{4*3}\). This gives us \(\sqrt[4]{x^{4*3}}\). Now, we can see that if 'n'=4 and 'a'=\(x^{3}\), then our expression becomes exactly like the radical property \(\sqrt[n]{a^n}\). Using this property, we simplify it to \(x^{3}\).

Step 3: Write the Final Answer

The simplified expression of \(\sqrt[4]{x^{12}}\) is \(x^{3}\) as we've reduced the index of the radical.

Key Concepts

Index of the Radical

When you encounter a radical expression, such as \(\sqrt[4]{x^{12}}\), the 'index of the radical' is an important element to identify. Specifically, the index is the small number just outside and above the root symbol. It tells us how many times a number, called the radicand, must be multiplied by itself to reach under the root. In our example, the index is 4. This means we are dealing with the fourth root. If no index is present, it is usually understood to be 2, representing a square root. This understanding is essential when simplifying radical expressions, as it determines how you will manipulate the radicand.

Radical Property

A key principle to understand when working with radicals is the 'radical property'. This rule states that \(\sqrt[n]{a^n} = a\). Effectively, if the exponent of the radicand matches the index, you can simplify the radical to the base itself. For instance, if you have \(a^{n}\) under an \(n\)-root, they effectively cancel each other out, resulting in \(a\). In our example, we applied this principle to \(\sqrt[4]{x^{12}}\) by reorganizing the radicand to the form \(x^{4*3}\), making it easier to see how the radical property facilitates simplification. The ability to apply this property correctly can simplify seemingly complex expressions.

Radicand

In a radical expression, the 'radicand' is the number or expression located inside the radical symbol. It is what you take the root of. For \(\sqrt[4]{x^{12}}\), the radicand is \(x^{12}\). Understanding the radicand is crucial because it determines how you will maneuver algebraically to simplify the expression. Breaking down \(x^{12}\) to \(x^{4*3}\) serves to align the radicand with the radical property efficiently. When simplifying radicals, insight into how the radicand interacts with both the index and any algebraic properties can lead to simpler and clearer results.

Exponents in Radicals

Exponents play a critical role when simplifying radicals. In the case of \(\sqrt[4]{x^{12}}\), the exponent '12' on \(x\) indicates how many times \(x\) is multiplied by itself. Working with exponents inside radicals requires dividing the exponent by the index to seek simplification. For \(x^{12}\), this means thinking of it as \((x^4)^3\) to easily apply the radical property, thus simplifying to \(x^3\). Understanding how to deconstruct and manipulate exponents is fundamental to solving radical expressions as it allows you to transform and simplify to reach the clearest and simplest form of the expression.