- Simplify the first term \(\sqrt[3]{54 a}\) as \(3 \sqrt[3]{2 a}\), which results in \(9 b^2 \sqrt[3]{2 a}\).
- Simplify the second term \(\sqrt[3]{2 a b^6}\) as \(b^2 \sqrt[3]{2 a}\), which results in \(3 b^2 \sqrt[3]{2 a}\).
- Add the terms together to get \(9 b^2 \sqrt[3]{2 a} + 3 b^2 \sqrt[3]{2 a} = 12 b^2 \sqrt[3]{2 a}\).
- The final result is \(\boxed{12 b^2(\sqrt[3]{2 a})}\).
Explanation
- Simplify the Expression
The goal is to simplify the algebraic expression \(3 b^2 \sqrt[3]{54 a} + 3 \sqrt[3]{2 a b^6}\). We will analyze each term individually to simplify the radical components.
For the first term, \(3 b^2 \sqrt[3]{54 a}\), note that \(54 = 27 \times 2 = 3^3 \times 2\). Therefore, we can simplify the cube root:
\[\sqrt[3]{54 a} = \sqrt[3]{27 \times 2 a} = \sqrt[3]{27} \times \sqrt[3]{2 a} = 3 \sqrt[3]{2 a}.\]
Substituting this back into the first term, we get:
\[3 b^2 (3 \sqrt[3]{2 a}) = 9 b^2 \sqrt[3]{2 a}.\]
For the second term, \(3 \sqrt[3]{2 a b^6}\), note that \(b^6 = (b^2)^3\). Therefore, the cube root of \(b^6\) is \(b^2\):
\[\sqrt[3]{2 a b^6} = \sqrt[3]{b^6} \times \sqrt[3]{2 a} = b^2 \sqrt[3]{2 a}.\]
Substituting this back into the second term, we get:
\[3 (b^2 \sqrt[3]{2 a}) = 3 b^2 \sqrt[3]{2 a}.\]
Finally, we combine the two simplified terms:
\[9 b^2 \sqrt[3]{2 a} + 3 b^2 \sqrt[3]{2 a} = (9 + 3) b^2 \sqrt[3]{2 a} = 12 b^2 \sqrt[3]{2 a}.\]
Comparing this result to the given options, we find it matches the option \(12 b^2 \sqrt[3]{2 a}\).
Examples
Simplifying radicals is essential in physics and engineering when dealing with formulas involving cube roots, such as calculating the volume of objects or analyzing complex wave patterns. By simplifying the expression \(3 b^2 \sqrt[3]{54 a} + 3 \sqrt[3]{2 a b^6}\) to \(12 b^2 \sqrt[3]{2 a}\), we reduce the computational complexity, making it easier to perform further calculations or plug in specific values for variables like \(a\) and \(b\). This type of algebraic manipulation ensures efficiency and accuracy in scientific modeling.