Problem 97 - College Algebra

Question

In Exercises \(95-102,\) simplify by reducing the index of the radical. $$\sqrt[3]{x^{6}}$$

Step-by-Step Solution

Answer

\(\sqrt[3]{x^{6}} = x^2\)

Step 1: Identify the index and the exponent

The index of the radical (the root to be taken) is \(3\), and the exponent of the variable \(x\) is \(6\).

Step 2: Simplify by reducing the exponent

The cube root of a variable raised to an exponent is equivalent to that variable raised to the exponent divided by the cube root's index. So, we divide the exponent of \(x\), which is \(6\), by the index, which is \(3\). This gives \(6/3 = 2\). Therefore, \(\sqrt[3]{x^{6}}\) simplifies to \(x^2\).

Key Concepts

Radical Index

The radical index is a crucial part of simplifying radicals. It defines the degree of the root you are trying to find. In the expression \(\sqrt[3]{x^6}\), the number 3 is the radical index, indicating that we need to find the cube root of the expression inside the radical.

Understanding the radical index helps you follow specific mathematical rules when working with radicals:

• A smaller index like 2 (square root) is commonly used, while a higher index indicates more complex roots like cube roots, fourth roots, and so on.
• The radical index directly influences how you simplify the expression beneath the radical. It tells you how many times a number or expression should be multiplied by itself to get the number under the radical sign.

Mastering the concept of the radical index will allow you to simplify expressions with confidence and ease.

Exponents

Exponents are another key component while simplifying radicals. They represent how many times a number, in this case, the variable \(x\), is multiplied by itself. In the context of \(\sqrt[3]{x^6}\), \(x^6\) signifies that \(x\) is used as a factor six times.

When simplifying, you often adjust the exponents according to the radical index:

• For the expression \(x^6\), the exponent 6 shows the power of the variable.
• By dividing the exponent by the radical's index, you reduce the complexity of the expression, converting it into a simpler form.

In this example, dividing 6 by the index 3 simplifies \(x^6\) to \(x^2\). So understanding how to manipulate exponents with radicals is vital for proper simplification.

Cube Root

The cube root is a specific type of radical that requires taking something to the third degree. The cube root of a number is what you multiply by itself three times to reach the original number inside the radical. For instance, in \(\sqrt[3]{x^6}\), you are asked to find the cube root of \(x^6\).

This is how cube roots behave:

• They help simplify expressions by breaking down the variables and constants into more manageable parts.
• The cube root of \(x^6\) means identifying a factor that, when used three times, returns \(x^6\).

By using cube root rules and dividing the exponent (6) by the radical index (3), we find our answer: \(x^2\). Simplifying cube roots is an essential skill in algebra that supports deeper understanding of numbers and their properties.

Variable Simplification

Variable simplification is the process of making an algebraic expression as simple as possible. In problems like \(\sqrt[3]{x^6}\), it involves reducing the complexity of radicals and exponents for easier interpretation.

Here's how simplification is done:

• First, identify the exponent and radical index to determine the simplification rule.
• Divide the exponent by the radical index to simplify the expression.
• Translate the complex radical form into a simpler multiplicative form, in this case, \(x^2\).

Effective variable simplification eliminates cumbersome calculations and presents clean, manageable results that are easy to understand and further manipulate if necessary. Applying this to \(\sqrt[3]{x^6}\), you reduce the expression to \(x^2\), demonstrating the power and utility of simplifying radicals and variables.