Problem 20 - Intermediate Algebra

Question

Add or subtract. $$ 3 \sqrt{7}-\sqrt[3]{x}+4 \sqrt{7}-3 \sqrt[3]{x} $$

Step-by-Step Solution

Answer

The simplified expression is \(7 \sqrt{7} - 4 \sqrt[3]{x}\).

Step 1: Identify Like Terms

In the expression \(3 \sqrt{7}-\sqrt[3]{x}+4 \sqrt{7}-3 \sqrt[3]{x}\), identify the terms that are alike. Like terms are terms that contain the same radicand and index. Here, \(3 \sqrt{7}\) and \(4 \sqrt{7}\) are like terms, as are \(-\sqrt[3]{x}\) and \(-3 \sqrt[3]{x}\).

Step 2: Add or Subtract Like Radicals

Add or subtract the coefficients of the like terms. For \(3 \sqrt{7}\) and \(4 \sqrt{7}\), sum the coefficients: \(3 + 4 = 7\), giving \(7 \sqrt{7}\). For \(-\sqrt[3]{x}\) and \(-3 \sqrt[3]{x}\), sum the coefficients: \(-1 - 3 = -4\), giving \(-4 \sqrt[3]{x}\).

Step 3: Combine Results

Combine the simplified terms from Step 2 to get the final result. This gives us \(7 \sqrt{7} - 4 \sqrt[3]{x}\).

Key Concepts

Like Terms

In algebra, the concept of like terms is essential when simplifying or combining expressions. Like terms are terms that contain the same variables raised to the same powers, including the same index and radicand in the context of radicals.
When simplifying radical expressions, identifying like terms helps simplify calculations:

• For example, in the radical expression \(3 \sqrt{7}\) and \(4 \sqrt{7}\), both share the radical \(\sqrt{7}\). Hence, they are like terms.
• Similarly, the terms \(-\sqrt[3]{x}\) and \(-3 \sqrt[3]{x}\) are like terms because both involve the cube root of \(x\).

This identification is crucial because it dictates which terms in an expression can be combined—terms without this alignment cannot be grouped together.

Addition and Subtraction of Radicals

The addition and subtraction of radicals may seem challenging, but it's all about managing like terms. These operations become straightforward once you identify which radicals can be combined. The rules are similar to basic addition and subtraction with numbers.
The key steps are:

• First, ensure you are working with like terms, as discussed previously.
• Next, focus on combining their coefficients while keeping the radical part (e.g., \(\sqrt{7}\) or \(\sqrt[3]{x}\)) unchanged.
• For \(3 \sqrt{7} + 4 \sqrt{7}\), add the coefficients: \(3 + 4\), which results in \(7 \sqrt{7}\).
• Similarly, for \(-\sqrt[3]{x} - 3 \sqrt[3]{x}\), the operation \(-1 - 3\) yields \(-4 \sqrt[3]{x}\).

The idea is simple: add or subtract the numerical coefficients, and keep the radical part unchanged unless further simplification is possible.

Algebraic Expressions

Algebraic expressions, like the one in this exercise, consist of variables, numbers, and operations. They are the foundation of algebra and encompass a wide variety of mathematical statements.
To work efficiently with algebraic expressions:

• Understand that each part of the expression (terms, coefficients, variables) plays a specific role.
• Acknowledge that simplifying expressions often involves operations like addition, subtraction, multiplication, or division.
• In the context of radicals, we extend these operations to include the simplification of terms, particularly by combining like terms as shown in our example: \(3 \sqrt{7}-\sqrt[3]{x}+4 \sqrt{7}-3 \sqrt[3]{x}\).
• Breaking down the expression step by step helps, converting a complex problem into manageable parts.

Grasp these basics, and handling such expressions becomes intuitive and less daunting.