Problem 68 - College Algebra with Modeling and Visualization

Question

Simplify the expression. Assume that all variables are positive. $$ 18 \sqrt[3]{3}+3 \sqrt[3]{3} $$

Step-by-Step Solution

Answer

The expression simplifies to \(21 \sqrt[3]{3}\).

Step 1: Identify Like Terms

The expression \(18 \sqrt[3]{3} + 3 \sqrt[3]{3}\) consists of two terms: \(18 \sqrt[3]{3}\) and \(3 \sqrt[3]{3}\). These terms are like terms because they both contain the cube root of 3, \(\sqrt[3]{3}\).

Step 2: Factor Out the Common Term

To combine the like terms, factor out the common cube root term, \(\sqrt[3]{3}\). This gives us \((18 + 3) \sqrt[3]{3}\).

Step 3: Simplify the Coefficient

Now, add the coefficients of the terms: \(18 + 3 = 21\). This results in \(21 \sqrt[3]{3}\).

Step 4: Write the Final Simplified Expression

The fully simplified form of the expression is \(21 \sqrt[3]{3}\).

Key Concepts

Like Terms

When simplifying expressions, especially those involving radicals or roots, identifying like terms is crucial. Like terms are terms that contain the same variables raised to the same power. In the expression given, both terms, \(18 \sqrt[3]{3}\) and \(3 \sqrt[3]{3}\), contain the cube root of 3, \(\sqrt[3]{3}\), which makes them like terms. This is similar to combining terms with the same variables in algebra, such as combining \(2x\) and \(5x\) because both have the variable \(x\).

• Like terms can be added or subtracted directly.
• Ensure that the variables and roots match exactly for terms to be considered like terms.
• Recognizing like terms simplifies the addition or subtraction process in radical expressions.

Factoring

Factoring is a method used to simplify expressions, particularly when dealing with like terms. In this context, factoring involves finding a common term (or a common factor) that can be taken out of each term in an expression. With the expression \(18 \sqrt[3]{3} + 3 \sqrt[3]{3}\), the cube root \(\sqrt[3]{3}\) is common in both terms. Factoring it out reveals the expression in a more manageable form: \((18 + 3) \sqrt[3]{3}\).

• Factoring is a step that makes it easier to simplify expressions.
• By taking out common factors, you reduce the complexity of expressions.
• This step is essential before final simplification, as it can significantly simplify the process.

Cube Roots

The cube root, represented as \(\sqrt[3]{...}\), is a type of radical expression. It indicates that you are looking for a number which, when multiplied by itself three times, gives the original number. For example, \(\sqrt[3]{3}\) refers to the number that must be multiplied three times to result in 3. Understanding cube roots is similar to understanding square roots, but involves three factors instead of two.

• Cube roots can simplify expressions by reducing the number of radicals.
• They are common in algebra problems, especially those involving volumes, as many geometric volume calculations require cube roots.
• In simplification tasks, cube roots often appear as common terms, making it essential to be comfortable manipulating them.