- Factor the first term's radical: \(\sqrt[3]{192ab^2} = \sqrt[3]{64 \times 3ab^2} = 4\sqrt[3]{3ab^2}\), leading to \(2ab(4\sqrt[3]{3ab^2}) = 8ab\sqrt[3]{3ab^2}\).
- Factor the second term's radical: \(\sqrt[3]{81a^4b^5} = \sqrt[3]{27a^3b^3 \times 3ab^2} = 3ab\sqrt[3]{3ab^2}\), leading to \(5(3ab\sqrt[3]{3ab^2}) = 15ab\sqrt[3]{3ab^2}\).
- Subtract the simplified terms: \(8ab\sqrt[3]{3ab^2} - 15ab\sqrt[3]{3ab^2} = -7ab\sqrt[3]{3ab^2}\).
- The final answer is \(\boxed{-7ab\sqrt[3]{3ab^2}}\).
Explanation
- Problem Analysis and Strategy
We are given the expression \(2ab\sqrt[3]{192ab^2} - 5\sqrt[3]{81a^4b^5}\). Our objective is to simplify this expression by using the properties of cube roots and factoring perfect cubes out of each radical. Recall that for any real numbers \(x, y\), we have \(\sqrt[3]{x \cdot y} = \sqrt[3]{x} \cdot \sqrt[3]{y}\). We will also use the property \(\sqrt[3]{x^3} = x\).
- Simplifying the First Term
First, let's focus on the first term: \(2ab\sqrt[3]{192ab^2}\). We need to find perfect cube factors in \(192ab^2\).
Note that \(192 = 64 \times 3 = 4^3 \times 3\).
So, \(\sqrt[3]{192ab^2} = \sqrt[3]{64 \times 3 \times a \times b^2} = \sqrt[3]{64} \times \sqrt[3]{3ab^2} = 4\sqrt[3]{3ab^2}\).
Substituting this back into the first term, we get:
\(2ab \times 4\sqrt[3]{3ab^2} = 8ab\sqrt[3]{3ab^2}\).
- Simplifying the Second Term
Now, let's consider the second term: \(5\sqrt[3]{81a^4b^5}\).
We look for perfect cube factors in \(81a^4b^5\).
We have \(81 = 27 \times 3 = 3^3 \times 3\).
Also, \(a^4 = a^3 \times a\) and \(b^5 = b^3 \times b^2\).
So, \(\sqrt[3]{81a^4b^5} = \sqrt[3]{27 \times 3 \times a^3 \times a \times b^3 \times b^2} = \sqrt[3]{27 \times a^3 \times b^3} \times \sqrt[3]{3ab^2} = 3 \times a \times b \sqrt[3]{3ab^2} = 3ab\sqrt[3]{3ab^2}\).
Substituting this back into the second term:
\(5 \times 3ab\sqrt[3]{3ab^2} = 15ab\sqrt[3]{3ab^2}\).
- Combining and Conclusion
Now we combine the simplified terms:
\(8ab\sqrt[3]{3ab^2} - 15ab\sqrt[3]{3ab^2}\).
Since both terms contain the same radical part \(\sqrt[3]{3ab^2}\) and the same variable part \(ab\), we can simply subtract the coefficients:
\((8 - 15)ab\sqrt[3]{3ab^2} = -7ab\sqrt[3]{3ab^2}\).
Comparing this with the given options, this matches the third option.
Examples
Simplifying radical expressions is vital in various fields, such as engineering and physics, where formulas involving roots frequently appear. For example, when calculating the volume of complex shapes or analyzing signals in electrical engineering, you often need to reduce radical expressions to make them easier to evaluate or compare. Mastering this skill allows for more efficient problem solving and better conceptual understanding of algebraic manipulation.