Problem 75 - College Algebra

Question

Add or subtract terms whenever possible. $$4 \sqrt[5]{2}+3 \sqrt[3]{2}$$

Step-by-Step Solution

Answer

The given expression cannot be simplified further, so the answer is \(4 \sqrt[5]{2} + 3 \sqrt[3]{2}\).

Step 1: Understanding the given terms

We are given two terms, \(4 \sqrt[5]{2}\) and \(3 \sqrt[3]{2}\). A radical term can be explained as follows: the number inside the radical sign is called the radicand, and the small number just outside and above the radical sign is the index. In the term \(4 \sqrt[5]{2}\), 2 is the radicand and 5 is the index, while in \(3 \sqrt[3]{2}\), 2 is the radicand and 3 is the index.

Step 2: Analyzing the possibility of addition or subtraction

The indices of both terms are not the same, and this tells us that no addition or subtraction operation can be performed. The terms with radical signs can only be added or subtracted only if their indices and radicands are the same.

Step 3: Providing the final form

Since the terms cannot be combined through addition or subtraction, the expression remains in its original form, which is \(4 \sqrt[5]{2} + 3 \sqrt[3]{2}\).

Key Concepts

Radicand

In radical expressions, the radicand is the number under the radical sign. In our exercise, both terms, \(4 \sqrt[5]{2}\) and \(3 \sqrt[3]{2}\), have the same radicand, which is 2. A radicand can be any real number, and it is the value that is being subjected to the root operation.

• Positive radicands result in real number outputs when the index is an odd number or when the index is a square root.
• Negative radicands, when paired with an even index other than 2, often lead to complex numbers because there is no real number whose square is negative.

Understanding the concept of radicands is essential, as it determines the type of number and the simplified form of a radical expression.

Index of a Radical

The index of a radical is a small number located just above and outside of the radical sign. It tells us which root of the radicand we are dealing with. In mathematical notation, if there is no index written, it defaults to 2, indicating a square root.

• For the term \(4 \sqrt[5]{2}\), the index is 5, meaning we're taking the fifth root of 2.
• For the term \(3 \sqrt[3]{2}\), the index is 3, meaning we're taking the cube root of 2.

The index plays a critical role in determining whether radical expressions can be simplified or combined. For instance, equal indices between radicals are a requisite for performing addition or subtraction operations. This is because the operations fundamentally depend on having like roots so that the calculations are consistent.

Addition and Subtraction of Radicals

Performing addition or subtraction with radical expressions requires specific conditions to be met. Particularly, radicals must have the same index and radicand to be added or subtracted.

• If both the radicand and index are the same across terms, then the terms are like radicals and can be combined directly by adding or subtracting their coefficients.
• For example, \(5 \sqrt{3} + 2 \sqrt{3} = 7 \sqrt{3}\) because both terms have a radicand of 3 and an index of 2 (square root).

However, if either the index or the radicand differs, as in the given expression \(4 \sqrt[5]{2} + 3 \sqrt[3]{2}\), the terms are unlike radicals and cannot be combined. Always check both the radicand and the index when examining expressions for potential simplification. This understanding is crucial to correctly manipulating more complex radical expressions.