Problem 69 - College Algebra

Question

In Exercises \(69-76,\) add or subtract terms whenever possible. $$4 \sqrt[5]{2}+3 \sqrt[5]{2}$$

Step-by-Step Solution

Answer

The result of adding the given terms is \(7\sqrt[5]{2}\).

Step 1: Identify Similar Terms

We first identify the similar terms in the given expression. Here, \(4 \sqrt[5]{2}\) and \(3 \sqrt[5]{2}\) are similar terms because they both contain the same base \(\sqrt[5]{2}\), also known as the fifth root of 2.

Step 2: Apply Addition

After identifying the similar terms, we apply the basic arithmetic operation, which is addition in this case. While adding, we only add the coefficients and carry the base as it is. The calculation will be: \(4 \sqrt[5]{2} + 3 \sqrt[5]{2} = (4+3) \sqrt[5]{2}\).

Step 3: Add Coefficients

Now, add the coefficients and simplify: \(4 + 3 = 7\). So, our simplified expression becomes \(7\sqrt[5]{2}\).

Key Concepts

Simplification of Expressions

Simplifying an expression is about making it shorter or easier to handle while retaining its original value. This process focuses on reducing expressions to their simplest form. For radical expressions like those involving roots, simplification typically means combining similar terms to create a tidier mathematical expression.

In the given exercise, each term contains the fifth root of 2, denoted as \(\sqrt[5]{2}\).

• A radical, represented by the root symbol, refers to a number that can produce a specific quantity when multiplied by itself a certain number of times.
• The goal is to combine similar radical terms, making the expression concise while maintaining equality.

While simplifying, it is crucial to observe whether terms share common radical bases, as seen with \(4\sqrt[5]{2}\) and \(3\sqrt[5]{2}\). Ensuring that these bases are identical is the first step towards efficient simplification.

Algebraic Operations

Algebra involves operations such as addition, subtraction, multiplication, and division. When working with algebraic expressions involving radicals, these operations become essential tools for manipulation.

In this exercise, we focus on addition, where combining radicals requires:

• Identifying like radicals: Radicals that have the same index and radicand can be added together.
• Adding coefficients only: Keep the radical part constant and add their numerical coefficients, like regular numbers.

For the expression \(4\sqrt[5]{2} + 3\sqrt[5]{2}\), we don't change the \(\sqrt[5]{2}\) part; instead, we add the 4 from the first term and 3 from the second, resulting in \(7\sqrt[5]{2}\). This technique is akin to collecting and adding similar variables in algebra.

Identifying Like Terms

Like terms have matching variables or radicals and can be combined together in an expression. Identifying these terms is a fundamental skill in simplifying algebraic expressions, as it helps reduce complexity and simplify calculations.

To identify like terms in an expression:

• Inspect the components after the coefficient, ensuring they are identical.
• This examination includes the variable, exponent, or, as in this exercise, the radical.

In the given example, both terms \(4\sqrt[5]{2}\) and \(3\sqrt[5]{2}\) contain \(\sqrt[5]{2}\). Hence, they qualify as like terms and can be added together. Understanding this allows us to combine them into one neat expression, \(7\sqrt[5]{2}\), which is the simplified form. Recognizing like terms reduces expression length and simplifies further calculations or interpretations.