Problem 6 - Algebra 2

Question

Add or subtract if possible. $$ 7 \sqrt[3]{x^{2}}-2 \sqrt[3]{x^{2}} $$

Step-by-Step Solution

Answer

The simplified form of the radical expression is \(5\sqrt[3]{x^{2}}\)

Step 1: Identify the like-radical terms

The like-radical terms are \(7 \sqrt[3]{x^{2}}\) and \(-2 \sqrt[3]{x^{2}}\), which means these terms have the same radicand (\(x^{2}\)) and index (3)

Step 2: Perform the operation

Subtract the coefficients of these terms: \(7 - 2 = 5\)

Step 3: Combine like-radical terms

The result after subtraction is \(5\sqrt[3]{x^{2}}\). This is your final answer

Key Concepts

Like Terms

In algebra, like terms are terms that share the same variables with the same exponents. This is crucial for combining terms, particularly when dealing with expressions that include radicals.
For example, in the expression given:

• \(7 \sqrt[3]{x^{2}}\)
• \(-2 \sqrt[3]{x^{2}}\)

both terms are considered like terms. This is because they have the same radicand, \(x^{2}\), and the same index of the radical, \(3\), making operations like addition and subtraction possible. Understanding like terms helps simplify expressions and solve equations more effectively. Once you recognize terms as alike, you simply proceed by manipulating their coefficients.

Radicand

The radicand is the number or expression inside the radical symbol. It is the value that the root is applied to. The structure of a radical expression is important to understand when manipulating these terms, especially in addition or subtraction.
In our given exercise:

• \(\sqrt[3]{x^{2}}\)

has a radicand of \(x^{2}\). This indicates we are considering the cube root of \(x^2\). Knowing the radicand allows you to determine if you can combine terms. If the radicands are different, then generally, you cannot easily add or subtract the terms.
However, if the radicand and the indices match, as they do in our case, the expressions can be combined by adjusting their coefficients.

Index of a Radical

The index of a radical is the small number written just outside the radical symbol's "check mark" portion. It signifies which root is being taken. For example, a cube root will have an index of 3, and a square root will usually have an implicit index of 2, even though it is often unwritten.
In the exercise:

• \(\sqrt[3]{x^{2}}\)

the index is 3. This identifies the expression as a cube root. The index must be the same among terms for them to be combined as like radicals.
If the indices are different, you cannot perform standard operations such as addition or subtraction on the entire terms. Recognizing and matching indices is essential for simplifying radical expressions.

Simplification of Radicals

Simplifying radicals involves reducing the expression to its simplest form. This often means combining like terms where possible and ensuring the radical components are fully simplified. In our problem:

• \(7 \sqrt[3]{x^{2}} - 2 \sqrt[3]{x^{2}}\)

we follow these steps:1. **Identify like terms**: Both terms have the same radicand \(x^2\) and index 3, allowing them to be combined.2. **Subtract coefficients**: Perform the subtraction on the coefficients, resulting in \(7 - 2 = 5\).3. **Recombine the radical expression**: The simplified radical is \(5\sqrt[3]{x^{2}}\).Simplifying radicals not only makes the expressions easier to work with but also allows for cleaner solutions in problems involving operations on radical expressions. By practicing simplification, one gains a deeper understanding of the underlying mathematical relationships.