Problem 6 - Elementary and Intermediate Algebra

Question

In the expression \(\sqrt[3]{27 x^{6}},\) the \(\quad\) is 3 and \(27 x^{6}\) is the ______________.

Step-by-Step Solution

Answer

27 x^6 is the radicand.

Step 1: Identify the Expression Structure

The given expression is a cube root function. The general form is \(\sqrt[3]{a}\), where \(a\) is the expression inside the cube root.

Step 2: Determine the Parts of the Expression

In \(\sqrt[3]{27 x^{6}}\), the '3' represents the index of the root, which is the degree of the root, meaning it is a cube root. The expression \(27 x^{6}\) is the content under the cube root.

Step 3: Define the Placeholders

The question asks for what each part of \(\sqrt[3]{27 x^{6}}\) represents. The '3' is the index of the root and \(27 x^{6}\) is the radicand, the expression that is under the root sign.

Key Concepts

Cube Root

The concept of a cube root revolves around finding a number that, when multiplied by itself three times, yields the original number you started with. It is a type of radical expression, where the radical sign is accompanied by an index of 3. This index indicates that the root is a cube root. To illustrate, the cube root of 8 is 2, because multiplying 2 by itself three times gives 8; in mathematical terms, this is represented as \(\sqrt[3]{8} = 2 \). Cube roots are particularly important in solving equations where the unknown variable is raised to the power of three.

When dealing with expressions like \(\sqrt[3]{27 x^{6}} \), you aim to determine what number or expression multiplied three times results in \(27 x^{6}\). Understanding this helps in simplifying complex algebraic expressions and solving higher degree polynomial equations.

Radicand

In any radical expression, the radicand is the term beneath the radical sign, which represents the value you are trying to find the root of. For a cube root, the radicand is found inside the cube root symbol. It is important to identify the radicand since it is the core element you work with to simplify or perform operations on a radical expression.

For example, in \(\sqrt[3]{27 x^{6}} \), the radicand is \(27 x^{6}\). This expression combines constants and variables, which can often both be components of a radicand. By understanding the structure of the radicand, you can break down the expression into more manageable parts that can be easily simplified or otherwise manipulated in algebraic processes.

Index of the Root

The index of the root, a fundamental component of radical expressions, determines the degree of the root being taken. In most mathematical expressions, it is located as a small numeral just above but to the left of the radical sign. This number, also called the degree of the root, tells you how many times you need to multiply the root to return to the original radicand value.

In expression \(\sqrt[3]{27 x^{6}} \), the number 3 is the index of the root, indicating a cube root. This means the expression is asking, "What multiplied by itself three times equals \(27 x^{6}\)?" Recognizing and understanding the index is vital for correctly simplifying radical expressions and manipulating them in equations. Mastery of using the index allows you to approach higher-degree roots, thus expanding your ability to solve a variety of mathematical problems.