Problem 6 - Intermediate Algebra

Question

Fill in the blanks. In the expression \(\sqrt[3]{27 x^{6}},\) the _____ is 3 and \(27 x^{6}\) is the _____.

Step-by-Step Solution

Answer

Index is 3, and radicand is \(27 x^6\).

Step 1: Identifying the Terms

The expression \( \sqrt[3]{27 x^{6}} \) is a cube root expression. In any radical expression of the form \( \sqrt[n]{a} \), \( n \) is known as the index, and \( a \) is the radicand.

Step 2: Recognizing the Index

The index of a root is the small number written just outside and above the radical sign. In \( \sqrt[3]{27 x^{6}} \), this number is 3.

Step 3: Recognizing the Radicand

The radicand is the quantity inside the radical symbol. In our expression, it is everything under the cube root: \( 27 x^{6} \).

Key Concepts

Understanding the Index of a Root

The index of a root is a crucial component in radical expressions. In mathematics, it denotes the degree of the root being taken. For example, in the expression \( \sqrt[n]{a} \), the letter \( n \) represents the index. It tells us how many times the radicand is multiplied by itself to achieve the original number.

In the specific expression \( \sqrt[3]{27 x^{6}} \), the number 3 is the index of the root. This means we are taking a cube root of the expression, which involves finding a number that, when multiplied by itself three times, results in the radicand.

• Higher index values indicate higher-degree roots such as fourth, fifth roots, and so on.
• If no index is visible, it is conventionally a square root with an implicit index of 2.

Recognizing the index helps understand the root's nature and perform accurate simplifications.

What is a Radicand?

The radicand is another vital component of a radical expression. It is the value or expression underneath the radical symbol that we are interested in.

In our example, \( \sqrt[3]{27 x^{6}} \), the radicand is \( 27 x^{6} \). This means the cube root applies to the combination of both the numerical part 27 and the variable expression \( x^{6} \).

• The radicand can include numbers, variables, or both.
• It determines what specific calculation the root is impacting.

Understanding the radicand allows for exact evaluation and simplification of radical expressions.

The Concept of Radical Expression

Radical expressions are essential in algebra and calculus, as they allow us to express roots in a structured form. A typical radical expression consists of a radical symbol, an index, and a radicand. Together, they describe how elements under the radical sign ( \( \sqrt{} \) or occasionally other symbols) should be handled.

For \( \sqrt[3]{27 x^{6}} \), it’s a cube root radical expression containing:

• An index (3), indicating the cube root.
• A radicand (\( 27 x^{6} \)), defining the specific terms under the radical sign.

By dissecting a radical expression into these elements, solving or simplifying complex algebraic problems becomes much more manageable. Acquainting yourself with radical expressions is a stepping stone to mastering more advanced mathematical techniques.