- Recognize that \(5\sqrt[3]{6c}\) and \(7\sqrt[3]{6c}\) are like terms with the common radical \(\sqrt[3]{6c}\).
- Add the coefficients \(5\) and \(7\) using the distributive property.
- Conclude that the sum is \(12\sqrt[3]{6c}\).
- The final answer is \(\boxed{12 \sqrt[3]{6c}}\).
Explanation
- Analyzing the Expression
The expression to be simplified is \(5\sqrt[3]{6c} + 7\sqrt[3]{6c}\), where \(c
eq 0\). We observe that both terms involve the same radical expression, \(\sqrt[3]{6c}\). In algebra, terms that contain the same radical part are called like terms. They can be added or subtracted by adding or subtracting their coefficients while keeping the radical part unchanged, following the distributive property of multiplication over addition: \(a \cdot X + b \cdot X = (a + b) \cdot X\). Here, let \(X = \sqrt[3]{6c}\), \(a = 5\), and \(b = 7\).
- Simplifying and Selecting the Correct Option
Applying the distributive property, we add the coefficients 5 and 7: \(5 + 7 = 12\). Therefore, the sum is \(12\sqrt[3]{6c}\). We compare this result with the provided options:
- \(35 \sqrt[3]{6c}\) (Incorrect)
- \(12 \sqrt[3]{12c}\) (Incorrect)
- \(12 \sqrt[3]{6c}\) (Correct)
- \(72c\) (Incorrect)
The simplified expression is equivalent to \(12\sqrt[3]{6c}\).
Examples
Simplifying radical expressions is a foundational skill in algebra, commonly used when solving geometric problems involving volumes or lengths. For example, if you are calculating the total length of two rods where one is \(5\sqrt[3]{6c}\) meters long and the other is \(7\sqrt[3]{6c}\) meters long, identifying them as like terms allows you to quickly determine that their total combined length is \(12\sqrt[3]{6c}\) meters. This skill streamlines complex algebraic manipulation in fields like physics and engineering.